Holt Algebra 1 5-1 Identifying Linear Functions Warm Up 1.Solve 2x – 3y = 12 for y. 3.The ratio of...

Holt Algebra 1

5-1 Identifying Linear Functions

Warm Up1.Solve 2x – 3y = 12 for y.

3.The ratio of red hair to brown hair in a class is 2:7. If one student is randomly chosen, what

is the probability that the student has red hair?

4. Suppose a letter of the alphabet is randomly chosen. What is the probability that it will be a letter that is found in the word Tennessee?

2. Graph for D: {–10, –5, 0, 5, 10}.

Holt Algebra 1


Identify linear functions and linear equations.

Graph linear functions that represent real-world situations and give their domain and range.

Objectives

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The graph represents a function because each domain value (x-value) is paired with exactly one range value (y-value). Notice that the graph is a straight line. A function whose graph forms a straight line is called a linear function.

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Example 1A: Identifying a Linear Function by Its Graph

Identify whether the graph represents a function. Explain. If the graph does represent a function, is the function linear?

Each domain value is paired with exactly one range value. The graph forms a line.

linear function

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Example 1B: Identifying a Linear Function by Its Graph


Each domain value is paired with exactly one range value. The graph is not a line.

not a linear function

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Example 1C: Identifying a Linear Function by Its Graph


The only domain value, –2, is paired with many different range values.

not a function

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Check It Out! Example 1b


Each domain value is paired with exactly one range value. The graph forms a line.

linear function

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Check It Out! Example 1c


Each domain value is not paired with exactly one range value.

not a function

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You can sometimes identify a linear function by looking a table or a list of ordered pairs. In a linear function, a constant change in x corresponds to a constant change in y.

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In this table, a constant change of +1 in x corresponds to constant change of –3 in y. These points satisfy a linear function.

The points from this table lie on a line.

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In this table, a constant change of +1 in x does not correspond to a constant change in y. These points do not satisfy a linear function.

The points from this table do not lie on a line.

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+4

+4

+4

+4

+3

+3

+3

+3

x y

0

4

8

12

16

–3

0

3

6

9

Example 2A: Identifying a Linear Function by Using Ordered Pairs

Tell whether the set of ordered pairs satisfies a linear function. Explain.

{(0, –3), (4, 0), (8, 3), (12, 6), (16, 9)}

Write the ordered pairs in a table.

Look for a pattern.

A constant change of +4 in x corresponds to a constant change of +3 in y.

These points satisfy a linear function.

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Example 2B: Identifying a Linear Function by Using Ordered Pairs

+2

+2

+2

+2

–12

–4

+4

+12

x y

–4

–2

0

2

4

13

1

–3

1

13

{(–4, 13), (–2, 1), (0, –3), (2, 1), (4, 13)}

Write the ordered pairs in a table.

Look for a pattern.

A constant change of 2 in x corresponds to different changes in y.

These points do not satisfy a linear function.

Tell whether the set of ordered pairs satisfies a linear function. Explain.

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Another way to determine whether a function is linear is to look at its equation. A function is linear if it is described by a linear equation. A linear equation is any equation that can be written in the standard form shown below.

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Notice that when a linear equation is written in standard form

• x and y both have exponents of 1.• x and y are not multiplied together.• x and y do not appear in denominators, exponents, or radical signs.

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For any two points, there is exactly one line that contains them both. This means you need only two ordered pairs to graph a line.

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Example 3A: Graphing Linear Functions

Tell whether the function is linear. If so, graph the function.

x = 2y + 4 Write the equation in standard form. Try to get both variables on the same side. Subtract 2y from both sides.

x = 2y + 4–2y –2y

x – 2y = 4The equation is in standard form

(A = 1, B = –2, C = 4).

The equation can be written in standard form, so the function is linear.

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Example 3A Continued

x = 2y + 4

To graph, choose three values of y, and use them to generate ordered pairs. (You only need two, but graphing three points is a good check.)

Plot the points and connect them with a straight line.

••

•y x = 2y + 4 (x, y)

0 x = 2(0) + 4 = 4 (4, 0)

–1 x = 2(–1) + 4 = 2 (2, –1)

–2 x = 2(–2) + 4 = 0 (0, –2)

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Example 3B: Graphing Linear Functions


xy = 4

This is not linear, because x and y are multiplied. It is not in standard form.

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Check It Out! Example 3a


y = 5x – 9 Write the equation in standard form. Try to get both variables on the same side. Subtract 5x from both sides.

The equation is in standard form (A = –5, B = 1, C = –9).

y = 5x – 9–5x –5x

–5x + y = – 9


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Check It Out! Example 3a Continued

To graph, choose three values of x, and use them to generate ordered pairs. (You only need two, but graphing three points is a good check.)

y = 5x – 9

Plot the points and connect them with a straight line.

•

•

•

x y = 5x – 9 (x, y)

0 y = 5(0) – 9 = –9 (0, –9)

1 y = 5(1) – 9 = –4 (1, –4)

2 y = 5(2) – 9 = 1 (2, 1)

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y = 12


The equation is in standard form (A = 0, B = 1, C = 12).

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Check It Out! Example 3b Continued

y = 12

y

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y = 2x

This is not linear, because x is an exponent.

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For linear functions whose graphs are not horizontal, the domain and range are all real numbers. However, in many real-world situations, the domain and range must be restricted. For example, some quantities cannot be negative, such as time.

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Sometimes domain and range are restricted even further to a set of points. For example, a quantity such as number of people can only be whole numbers. When this happens, the graph is not actually connected because every point on the line is not a solution. However, you may see these graphs shown connected to indicate that the linear pattern, or trend, continues.

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Check It Out! Example 4

What if…? At a salon, Sue can rent a station for $10.00 per day plus $3.00 per manicure. The amount she would pay each day is given by f(x) = 3x + 10, where x is the number of manicures. Graph this function and give its domain and range.

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Check It Out! Example 4 Continued

The number of manicures must be a whole number, so the domain is {0, 1, 2, 3, …}. The range is {10.00, 13.00, 16.00, 19.00, …}.

Choose several values of x and make a table of ordered pairs.

f(x) = 3x + 10x

1

3

5

f(1) = 3(1) + 10 = 13

f(3) = 3(3) + 10 = 19

f(5) = 3(5) + 10 = 25

f(0) = 3(0) + 10 = 100

2 f(2) = 3(2) + 10 = 16

4 f(4) = 3(4) + 10 = 22

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Graph the ordered pairs.

The individual points are solutions in this situation. The line shows that the trend continues.

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5-2 Using Intercepts

Warm-Up

Tell whether each set of ordered pairs satisfies a linear function. Explain.

1. {(–3, 10), (–1, 9), (1, 7), (3, 4), (5, 0)}

2. {(3, 4), (5, 7), (7, 10), (9, 13), (11, 16)}

Tell whether each function is linear. If so, graph the function.

3. y = 3 – 2x 4. 3y = 12

5. The cost of a can of iced-tea mix at Save More Grocery is $4.75. The function f(x) = 4.75x gives the cost of x cans of iced-tea mix. Graph this function and give its domain and range.

Holt Algebra 1


Find x- and y-intercepts and interpret their meanings in real-world situations.

Use x- and y-intercepts to graph lines.

Objectives

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The y-intercept is the y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0.

The x-intercept is the x-coordinate of the point where the graph intersects the x-axis. The y-coordinate of this point is always 0.

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Example 1A: Finding Intercepts

Find the x- and y-intercepts.

The graph intersects the y-axis at (0, 1).

The y-intercept is 1.

The graph intersects the x-axis at (–2, 0).

The x-intercept is –2.

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Example 1B: Finding Intercepts

5x – 2y = 10

5x – 2y = 10

5x – 2(0) = 10

5x – 0 = 10

5x = 10

To find the y-intercept, replace x with 0 and solve for y.

To find the x-intercept, replace y with 0 and solve for x.

x = 2The x-intercept is 2.

5x – 2y = 10

5(0) – 2y = 10

0 – 2y = 10 – 2y = 10

y = –5The y-intercept is –5.

Find the x- and y-intercepts.

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Example 2: Sports Application

Trish can run the 200 m dash in 25 s. The function f(x) = 200 – 8x gives the distance remaining to be run after x seconds. Graph this function and find the intercepts. What does each intercept represent?

Neither time nor distance can be negative, so choose several nonnegative values for x. Use the function to generate ordered pairs.

f(x) = 200 – 8x 200

250 5 10 20x

160 120 40 0

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Graph the ordered pairs. Connect the points with a line.

x-intercept: 25. This is the time it takes Trish to finish the race, or when the distance remaining is 0.

y-intercept: 200. This is the number of meters Trish has to run at the start of the race.

Example 2 Continued

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Check It Out! Example 2aThe school sells pens for $2.00 and notebooks for $3.00. The equation 2x + 3y = 60 describes the number of pens x and notebooks y that you can buy for $60.

Graph the function and find its intercepts.

x 0 15 30

20 10 0

Neither pens nor notebooks can be negative, so choose several nonnegative values for x. Use the function to generate ordered pairs.

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Check It Out! Example 2a ContinuedThe school sells pens for $2.00 and notebooks for $3.00. The equation 2x + 3y = 60 describes the number of pens x and notebooks y that you can buy for $60.

Graph the function and find its intercepts.

x-intercept: 30; y-intercept: 20

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Check It Out! Example 2bThe school sells pens for $2.00 and notebooks for $3.00. The equation 2x + 3y = 60 describes the number of pens x and notebooks y that you can buy for $60.

x-intercept: 30. This is the number of pens that can be purchased if no notebooks are purchased.

y-intercept: 20. This is the number of notebooks that can be purchased if no pens are purchased.

What does each intercept represent?

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Remember, to graph a linear function, you need to plot only two ordered pairs. It is often simplest to find the ordered pairs that contain the intercepts.

Helpful Hint

You can use a third point to check your line. Either choose a point from your graph and check it in the equation, or use the equation to generate a point and check that it is on your graph.

Holt Algebra 1


Example 3A: Graphing Linear Equations by Using Intercepts

Use intercepts to graph the line described by the equation.3x – 7y = 21

Step 1 Find the intercepts.x-intercept: y-intercept:

3x – 7y = 21

3x – 7(0) = 21

3x = 21

x = 7

3x – 7y = 21

3(0) – 7y = 21

–7y = 21

y = –3

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Step 2 Graph the line.

Plot (7, 0) and (0, –3).

Connect with a straight line.


Use intercepts to graph the line described by the equation.3x – 7y = 21

x

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Use intercepts to graph the line described by the equation.

Step 1 Write the equation in standard form.


3y = x – 6

–x + 3y = –6

Multiply both sides 3, to clear the fraction.

Write the equation in standard form.

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Step 2 Find the intercepts.x-intercept: y-intercept:

–x + 3y = –6

–x + 3(0) = –6

–x = –6

x = 6

–x + 3y = –6

–(0) + 3y = –6

3y = –6

y = –2



–x + 3y = –6

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Plot (6, 0) and (0, –2).

Connect with a straight line.


–x + 3y = –6

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5-3 Rate of Change and Slope

Warm-Up1. An amateur filmmaker has $6000 to make a film

that costs $75/h to produce. The function f(x) = 6000 – 75x gives the amount of money left to make the film after x hours of production. Graph this function and find the intercepts. What does each intercept represent?

2. Use intercepts to graph the line described by

Holt Algebra 1


Find rates of change and slopes.

Relate a constant rate of change to the slope of a line.

Objectives

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A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.

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Example 1: Application

The table shows the average temperature (°F) for five months in a certain city. Find the rate of change for each time period. During which time period did the temperature increase at the fastest rate?

Step 1 Identify the dependent and independent variables.

dependent: temperature independent: month

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Step 2 Find the rates of change.

Example 1 Continued

The temperature increased at the greatest rate from month 5 to month 7.

3 to 5

5 to 7

7 to 8

2 to 3

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Example 2: Finding Rates of Change from a Graph

Graph the data from Example 1 and show the rates of change.

Graph the ordered pairs. The vertical segments show the changes in the dependent variable, and the horizontal segments show the changes in the independent variable.

Notice that the greatest rate of change is represented by the steepest of the red line segments.

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Example 2 Continued

Graph the data from Example 1 and show the rates of change.

Also notice that between months 2 to 3, when the balance did not change, the line segment is horizontal.

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If all of the connected segments have the same rate of change, then they all have the same steepness and together form a straight line. The constant rate of change of a line is called the slope of the line.

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Example 3: Finding Slope

Find the slope of the line.

Begin at one point and count vertically to fine the rise.

Then count horizontally to the second point to find the run.

It does not matter which point you start with. The slope is the same.

(3, 2)

(–6, 5)

•

•

Rise 3

Run –9

Rise –3

Run 9

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Example 4: Finding Slopes of Horizontal and Vertical Lines

Find the slope of each line.

You cannot divide by 0

The slope is undefined. The slope is 0.

A. B.

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As shown in the previous examples, slope can be positive, negative, zero or undefined. You can tell which of these is the case by looking at a graph of a line–you do not need to calculate the slope.

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Tell whether the slope of each line is positive, negative, zero or undefined.

a. b.

The line rises from left to right. The slope is positive.

The line is vertical.

The slope is undefined.

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5-4 The Slope FormulaWarm-Up

Name each of the following.

1. The table shows the number of bikes made by a company for certain years. Find the rate of change for each time period. During which time period did the number of bikes increase at the fastest rate?

Find the slope of each line.

2. 3.

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5-4 The Slope Formula

Find slope by using the slope formula.

Objective

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Example 1: Finding Slope by Using the Slope Formula

Find the slope of the line that contains (2, 5) and (8, 1).

The slope of the line that contains (2, 5) and (8, 1)

is .

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Find the slope of the line that contains (–2, –2) and (7, –2).


The slope of the line that contains (–2, –2) and (7, –2) is 0.

= 0

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Find the slope of the line that contains and


The slope of the line that contains and

is 2.

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Sometimes you are not given two points to use in the formula. You might have to choose two points from a graph or a table.

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Example 2A: Finding Slope from Graphs and Tables

The graph shows a linear relationship. Find the slope.

Let (0, 2) be (x1, y1) and (–2, –2) be (x2, y2).

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Example 2B: Finding Slope from Graphs and Tables

The table shows a linear relationship. Find the slope.

Step 1 Choose any two points from the table. Let (0, 1) be (x1, y1) and (–2, 5) be (x2, y2).Step 2 Use the slope formula.

The slope equals −2

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Remember that slope is a rate of change. In real-world problems, finding the slope can give you information about how a quantity is changing.

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The graph shows the average electricity costs (in dollars) for operating a refrigerator for several months. Find the slope of the line. Then tell what the slope represents.

Step 1 Use the slope formula.

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Example 3 Continued

Step 2 Tell what the slope represents.

In this situation y represents the cost of electricity and x represents time.

So slope represents in units of

.

A slope of 6 mean the cost of running the refrigerator is a rate of 6 dollars per month.

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If you know the equation that describes a line, you can find its slope by using any two ordered-pair solutions. It is often easiest to use the ordered pairs that contain the intercepts.

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Example 4: Finding Slope from an Equation

Find the slope of the line described by 4x – 2y = 16.

Step 1 Find the x-intercept. Step 2 Find the y-intercept.

4x – 2y = 16

4x = 16

x = 4Step 3 The line contains (4, 0) and (0, –8). Use the

slope formula.

4x – 2y = 16

–2y = 16

y = –8

4x – 2(0) = 16 Let y = 0. 4(0) – 2y = 16 Let x = 0.

Holt Geometry

1-6 Midpoint and Distance in the Coordinate Plane

5 Minute Warm-Up

Find the area and perimeter/circumference of each figure.

1. 2.

3.

4. circle with radius 2 cm

5. circle with diameter 12 ft

Holt Geometry


Develop and apply the formula for midpoint.

Use the Distance Formula and the Pythagorean Theorem to find the distance between two points.

Objectives

Holt Geometry


A coordinate plane is a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis) . The location, or coordinates, of a point are given by an ordered pair (x, y).

Holt Geometry


You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.

Holt Geometry


Holt Geometry


Example 1: Finding the Coordinates of a Midpoint

Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7).

= (–5, 5)

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Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).

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Example 2: Finding the Coordinates of an Endpoint

M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y.

Step 1 Let the coordinates of Y equal (x, y).

Step 2 Use the Midpoint Formula:

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Example 2 Continued

Step 3 Find the x-coordinate.

Set the coordinates equal.

Multiply both sides by 2.

12 = 2 + x Simplify.

– 2 –2

10 = x

Subtract.

Simplify.

2 = 7 + y– 7 –7

–5 = y

The coordinates of Y are (10, –5).

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Step 3 Find the x-coordinate.

Set the coordinates equal.

Multiply both sides by 2.

–2 = –6 + x Simplify.

+ 6 +6

4 = x

Add.

Simplify.

2 = –1 + y+ 1 + 1

3 = y

The coordinates of T are (4, 3).

Holt Geometry


The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between two points in a coordinate plane.

Holt Geometry


Example 3: Using the Distance Formula

Find FG and JK. Then determine whether FG JK.

Step 1 Find the coordinates of each point.

F(1, 2), G(5, 5), J(–4, 0), K(–1, –3)

Holt Geometry


Example 3 Continued

Step 2 Use the Distance Formula.

Holt Geometry



Step 2 Use the Distance Formula.

Holt Geometry


You can also use the Pythagorean Theorem to find the distance between two points in a coordinate plane. You will learn more about the Pythagorean Theorem in Chapter 5.

In a right triangle, the two sides that form the right angle are the legs. The side across from the right angle that stretches from one leg to the other is the hypotenuse. In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c.

Holt Geometry


Holt Geometry


Example 4: Finding Distances in the Coordinate Plane

Use the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from D(3, 4) to E(–2, –5).

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Example 4 Continued

Method 1Use the Distance Formula. Substitute thevalues for the coordinates of D and E into theDistance Formula.

Holt Geometry


Method 2Use the Pythagorean Theorem. Count the units for sides a and b.

Example 4 Continued

a = 5 and b = 9.

c2 = a2 + b2

= 52 + 92

= 25 + 81

= 106

c = 10.3

Holt Geometry


A player throws the ball from first base to a point located between third base and home plate and 10 feet from third base. What is the distance of the throw, to the nearest tenth?

Example 5: Sports Application

Holt Geometry


Set up the field on a coordinate plane so that home plate H is at the origin, first base F has coordinates (90, 0), second base S has coordinates (90, 90), and third base T has coordinates (0, 90).

The target point P of the throw has coordinates (0, 80). The distance of the throw is FP.

Example 5 Continued

Holt Algebra 1

5-5 Direct Variation

Warm-Up

1. Find the slope of the line that contains (5, 3)

and (–1, 4).

2. Find the slope of the line. Then tell what the slope represents.

50; speed of bus is 50 mi/h

3. Find the slope of the line described by x + 2y = 8.

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Identify, write, and graph direct variation.

Objective

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A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef needs 1 cup of rice for every 5 servings.

The equation y = 5x describes this relationship. In this relationship, the number of servings varies directly with the number of cups of rice.

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A direct variation is a special type of linear relationship that can be written in the form y = kx, where k is a nonzero constant called the constant of variation.

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Example 1A: Identifying Direct Variations from Equations

Tell whether the equation represents a direct variation. If so, identify the constant of variation. y = 3x

This equation represents a direct variation because it is in the form of y = kx. The constant of variation is 3.

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3x + y = 8 Solve the equation for y.Since 3x is added to y, subtract 3x

from both sides.–3x –3x

y = –3x + 8

This equation is not a direct variation because it cannot be written in the form y = kx.

Example 1B: Identifying Direct Variations from Equations

Tell whether the equation represents a direct variation. If so, identify the constant of variation.

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–4x + 3y = 0 Solve the equation for y.Since –4x is added to 3y, add 4x

to both sides.

+4x +4x3y = 4x

This equation represents a direct variation because it is in the form of y = kx. The constant of variation is .

Since y is multiplied by 3, divide both sides by 3.

Example 1C: Identifying Direct Variations from Equations

Tell whether the equation represents a direct variation. If so, identify the constant of variation.

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What happens if you solve y = kx for k?

y = kx

So, in a direct variation, the ratio is equal to the constant of variation. Another way to identify a direct variation is to check whether is the same for each ordered pair (except where x = 0).

Divide both sides by x (x ≠ 0).

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Tell whether the relationship is a direct variation. Explain.

Method 2 Find for each ordered pair.

This is a direct variation because is the same for each ordered pair.

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Method 2 Find for each ordered pair.

This is not direct variation because is the not the same for all ordered pairs.

Example 2B Continued

Tell whether the relationship is a direct variation. Explain.

…

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Example 3: Writing and Solving Direct Variation Equations

The value of y varies directly with x, and y = 3, when x = 9. Find y when x = 21.

Method 1 Find the value of k and then write the equation.

y = kx Write the equation for a direct variation.

3 = k(9) Substitute 3 for y and 9 for x. Solve for k.

Since k is multiplied by 9, divide both sides by 9.

The equation is y = x. When x = 21, y = (21) = 7.

Holt Algebra 1



The value of y varies directly with x, and y = 4.5 when x = 0.5. Find y when x = 10.

Method 1 Find the value of k and then write the equation.

y = kx Write the equation for a direct variation.

4.5 = k(0.5) Substitute 4.5 for y and 0.5 for x. Solve for k.

Since k is multiplied by 0.5, divide both sides by 0.5.

The equation is y = 9x. When x = 10, y = 9(10) = 90.

9 = k

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Lesson Quiz: Part ITell whether each equation represents a direct variation. If so, identify the constant of variation.

1. 2y = 6x yes; 3

2. 3x = 4y – 7 no

Tell whether each relationship is a direct variation. Explain.

3. 4.

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Lesson Quiz: Part II

5. The value of y varies directly with x, and y = –8 when x = 20. Find y when x = –4. 1.6

6. Apples cost $0.80 per pound. The equation y = 0.8x describes the cost y of x pounds of apples. Graph this direct variation.

2

4

6

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5-6 Slope-Intercept Form

Write a linear equation in slope-intercept form.Graph a line using slope-intercept form.

Objectives

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Example 1A: Graphing by Using Slope and y-intercept

Graph the line given the slope and y-intercept.

y intercept = 4

Step 1 The y-intercept is 4, so the line contains (0, 4). Plot (0, 4).

Step 3 Draw the line through the two points.

•

••

•

Run = 5

Rise = –2

Step 2 Slope = Count

2 units down and 5 units right

from (0, 4) and plot another point.

y

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Graph the line given the slope and y-intercept.

slope = 2, y-intercept = –3

Step 1 The y-intercept is –3, so the line contains (0, –3). Plot (0, –3).

•

•

Step 3 Draw the line through the two points.

Rise = 2

Run = 1

Step 2 Slope =

Count 2 units up and 1 unit right

from (0, –3) and plot another

point.

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If you know the slope of a line and the y-intercept, you can write an equation that describes the line.

Step 1 If a line has a slope of 2 and the y-intercept is 3, then m = 2 and (0, 3) is on the line. Substitute these values into the slope formula.

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Step 2 Solve for y:

Simplify the denominator.

• •

2x = y – 3+3 +3

2x + 3 = y, or y = 2x + 3

Multiply both sides by x.

Add 3 to both sides.

Holt Algebra 1


Any linear equation can be written in slope-intercept form by solving for y and simplifying. In this form, you can immediately see the slope and y-intercept. Also, you can quickly graph a line when the equation is written in slope-intercept form.

Holt Algebra 1


Example 2A: Writing linear Equations in Slope-Intercept Form

Write the equation that describes the line in slope-intercept form.

slope = ; y-intercept = 4

y = mx + b

y = x + 4

Substitute the given values for m and b.

Simplify if necessary.

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slope = –9; y-intercept =

y = mx + b Substitute the given values for m and b.

Simplify if necessary.

Example 2B: Writing linear Equations in Slope-Intercept Form


y = –9x +

Holt Algebra 1


slope = 2; (3, 4) is on the line

y = mx + b

Substitute 2 for m, 3 for x, and 4 for y.

4 = 2(3) + b

Step 1 Find the y-intercept.

–2 = b

4 = 6 + b–6 –6

Write the slope-intercept form.

Solve for b. Since 6 is added to b, subtract 6 from both sides to undo the addition.

Example 2E: Writing linear Equations in Slope-Intercept Form


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Example 2E Continued

Step 2 Write the equation.

y = mx + b Write the slope-intercept form.

Substitute 2 for m, and –2 for b.y = 2x + (–2)

y = 2x – 2

slope = 2; (3, 4) is on the line


Holt Algebra 1


Example 3B: Using Slope-Intercept Form to Graph

Write the equation in slope-intercept form. Then graph the line described by the equation.

2y + 3x = 6

Step 1 Write the equation in slope-intercept form by solving for y.

2y + 3x = 6 –3x –3x2y = –3x + 6

Subtract 3x from both sides.

Since y is multiplied by 2, divide both sides by 2.

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Example 3B Continued


slope: m =

y-intercept: b = 3Plot (0, 3).

• Count 3 units down and 2 units right and plot another point.

• Draw the line connecting the two points.

•

•


is in the form y = mx + b.

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is in the form y = mx + b.

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y-intercept: b = 0

Step 1 Plot (0, 0).

Step 2 Count 2 units up and 3 units right and plot another point.

Step 3 Draw the line connecting the two points.

•

•y = x + 0 is in the form y = mx + b.

slope:


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y = –4

Step 1 Plot (0, –4).

y = –4 is in the form y = mx + b.

y-intercept: b = –4

slope: m = 0 = = 0

Since the slope is 0, the line will be a horizontal at y = –4.

•


Holt Algebra 1



A closet organizer charges a $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below.

Holt Algebra 1



a. Write an equation that represents the cost as a function of the number of hours.

Cost is $30for each

hour plus $100

y = 30 •x + 100

An equation is y = 30x + 100.

A closet organizer charges $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below.

Holt Algebra 1


Example 4 ContinuedA closet organizer charges $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below.b. Identify the slope and y-intercept and describe

their meanings.

The y-intercept is 100. This is the cost for 0 hours, or the initial fee of $100. The slope is 30. This is the rate of change of the cost: $30 per hour.

c. Find the cost if the organizer works 12 hrs.

y = 30x + 100= 30(12) + 100 = 460

Substitute 12 for x in the equation

The cost of the organizer for 12 hours is $460.

Holt Algebra 1

5-7 Point-Slope Form

Warm-Up

Write the equation that describes each line in the slope-intercept form.

1. slope = 3, y-intercept = –2

2. slope = 0, y-intercept =

3. slope = , (2, 7) is on the line

Write each equation in slope-intercept form. Then graph the line described by the equation.4. 6x + 2y = 10 5. x – y = 6

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Graph a line and write a linear equation using point-slope form.

Write a linear equation given two points.

Objectives

Holt Algebra 1


In lesson 5-6 you saw that if you know the slope of a line and the y-intercept, you can graph the line. You can also graph a line if you know its slope and any point on the line.

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•

•2

Example 1A: Using Slope and a Point to Graph

Graph the line with the given slope that contains the given point.

slope = 2; (3, 1)

Step 1 Plot (3, 1).

Step 2 Use the slope to move from (3, 1) to another point.

Move 2 units up and 1 unit right and plot another point.

Step 3 Draw the line connecting the two points.

1

(3, 1)

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Example 1C: Using Slope and a Point to Graph

Graph the line with the given slope that contains the given point.

slope = 0; (4, –3)

A line with a slope of 0 is horizontal. Draw the horizontal line through (4, –3).

(4, –3)

•

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If you know the slope and any point on the line, you can write an equation of the line by using the slope formula. For example, suppose a line has a slope of 3 and contains (2, 1). Let (x, y) be any other point on the line.

3(x – 2) = y – 1

y – 1 = 3(x – 2)

Slope formula

Substitute into the slope formula.

Multiply both sides by (x – 2).

Simplify.

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Holt Algebra 1


Example 2: Writing Linear Equations in Point-Slope Form

Write an equation in point-slope form for the line with the given slope that contains the given point.A. B. C.

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5-7 Point-Slope FormExample 3: Writing Linear Equations in Slope-Intercept

Form

Write an equation in slope-intercept form for the line with slope 3 that contains (–1, 4).

Step 1 Write the equation in point-slope form:

y – 4 = 3[x – (–1)]Step 2 Write the equation in slope-intercept form by

solving for y.

y – 4 = 3(x + 1)Rewrite subtraction of negative

numbers as addition.Distribute 3 on the right side.y – 4 = 3x + 3

+ 4 + 4

y = 3x + 7


y – y1 = m(x – x1)

Holt Algebra 1


Example 4A: Using Two Points to Write an Equation

Write an equation in slope-intercept form for the line through the two points.

(2, –3) and (4, 1)

Step 1 Find the slope.

Step 2 Substitute the slope and one of the points into the point-slope form.

Choose (2, –3).

y – y1 = m(x – x1)

y – (–3) = 2(x – 2)

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Step 3 Write the equation in slope-intercept form.

y = 2x – 7

–3 –3


Write an equation in slope-intercept form for the line through the two points.

(2, –3) and (4, 1)

y + 3 = 2(x – 2)

y + 3 = 2x – 4

Holt Algebra 1


Example 5: Problem-Solving Application

The cost to stain a deck is a linear function of the deck’s area. The cost to stain 100, 250, 400 square feet are shown in the table. Write an equation in slope-intercept form that represents the function. Then find the cost to stain a deck whose area is 75 square feet.

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Understand the Problem1

• The answer will have two parts—an equation in slope-intercept form and the cost to stain an area of 75 square feet.

• The ordered pairs given in the table—(100, 150), (250, 337.50), (400, 525)—satisfy the equation.

Example 5 Continued

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2 Make a Plan

You can use two of the ordered pairs to find the slope. Then use point-slope form to write the equation. Finally, write the equation in slope-intercept form.

Example 5 Continued

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Solve3

Step 1 Choose any two ordered pairs from the table to find the slope.

Use (100, 150) and (400, 525).

Step 2 Substitute the slope and any ordered pair from the table into the point-slope form.

y – 150 = 1.25(x – 100) Use (100, 150).

Example 5 Continued

y – y1 = m(x – x1)

Holt Algebra 1


Step 3 Write the equation in slope-intercept form by solving for y.

y – 150 = 1.25(x – 100)

y – 150 = 1.25x – 125 Distribute 1.25.

y = 1.25x + 25 Add 150 to both sides.

Step 4 Find the cost to stain an area of 75 sq. ft.y = 1.25x + 25

y = 1.25(75) + 25 = 118.75

The cost of staining 75 sq. ft. is $118.75.

Example 5 Continued

Holt Algebra 1

5-8 Slopes of Parallel and Perpendicular Lines

Warm-UpWrite an equation in point-slope form and slope-intercept form for the line through the two points.

1. (–1, 7) and (2, 1)

2. The cost to take a taxi from

the airport is a linear function of

the distance driven. The cost for

5, 10, and 20 miles are shown in the

table. Write an equation in

slope-intercept form that represents the function.

Holt Algebra 1


Identify and graph parallel and perpendicular lines.

Write equations to describe lines parallel or perpendicular to a given line.

Objectives

Holt Algebra 1


To sell at a particular farmers’ market for a year, there is a $100 membership fee. Then you pay $3 for each hour that you sell at the market. However, if you were a member the previous year, the membership fee is reduced to $50.

• The red line shows the total cost if you are a new member.

• The blue line shows the total cost if you are a returning member.

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These two lines are parallel. Parallel lines are lines in the same plane that have no points in common. In other words, they do not intersect.

Holt Algebra 1


Holt Algebra 1


Example 1A: Identifying Parallel Lines

Identify which lines are parallel.

The lines described by

and both have slope .

These lines are parallel. The lines

described by y = x and y = x + 1

both have slope 1. These lines

are parallel.

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Write all equations in slope-intercept form to determine the slope.

Slope-intercept formy = 3x

Slope-intercept form

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Write all equations in slope-intercept form to determine the slope.

–3x + 4y = 32+3x +3x

4y = 3x + 32

y – 1 = 3(x + 2)y – 1 = 3x + 6

+ 1 + 1

y = 3x + 7

Holt Algebra 1



The lines described by

–3x + 4y = 32 and y = + 8

have the same slope, but they

are not parallel lines. They are

the same line.

–3x + 4y = 32

y – 1 = 3(x + 2)

y = 3x

The lines described by y = 3x and y – 1 = 3(x + 2) represent parallel lines. They each have slope 3.

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Perpendicular lines are lines that intersect to form right angles (90°).

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Holt Algebra 1


Identify which lines are perpendicular: y = 3; x = –2; y = 3x; .

The graph described by y = 3 is a horizontal line, and the graph described by x = –2 is a vertical line. These lines are perpendicular.

y = 3x = –2

y =3x

The slope of the line described by y = 3x is 3. The slope of the line described by is .

Example 3: Identifying Perpendicular Lines

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Identify which lines are perpendicular: y = 3; x = –2; y = 3x; .

y = 3x = –2

y =3x

Example 3 Continued

These lines are perpendicular because the product of their slopes is –1.

Holt Algebra 1


Example 4: Geometry Application

Show that ABC is a right triangle.

slope of

slope of

Therefore, ABC is a right triangle because it contains a right angle.

If ABC is a right triangle, AB will be perpendicular to AC.

AB is perpendicular to AC because

Holt Algebra 1


Example 5A: Writing Equations of Parallel and Perpendicular Lines

Write an equation in slope-intercept form for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8.

Step 1 Find the slope of the line.

y = 3x + 8 The slope is 3.

The parallel line also has a slope of 3.

Step 2 Write the equation in point-slope form.

Use the point-slope form.y – y1 = m(x – x1)

y – 10 = 3(x – 4) Substitute 3 for m, 4 for x1, and 10 for y1.

Holt Algebra 1



Write an equation in slope-intercept form for the line that passes through (4, 10) and is parallel to the line described by y = 3x + 8.


y – 10 = 3(x – 4)

y – 10 = 3x – 12

y = 3x – 2

Distribute 3 on the right side.


Holt Algebra 1


Helpful Hint

If you know the slope of a line, the slope of a perpendicular line will be the "opposite reciprocal.”

Holt Algebra 1



Write an equation in slope-intercept form for the line that passes through (5, 7) and is parallel to the line described by y = x – 6.

Step 1 Find the slope of the line.

Step 2 Write the equation in point-slope form.

Use the point-slope form.

y = x –6 The slope is .

The parallel line also has a slope of .

y – y1 = m(x – x1)

Holt Algebra 1



Write an equation in slope-intercept form for the line that passes through (5, 7) and is parallel to the line described by y = x – 6.



Distribute on the right side.

Holt McDougal Algebra 1

5-10 Transforming Linear Functions

Warm-Up

Write an equation in slope-intercept form for the line described.

1. contains the point (8, –12) and is parallel to

2. contains the point (4, –3) and is perpendicular

to y = 4x + 5

3. Show that WXYZ is a rectangle.



Describe how changing slope and y-intercept affect the graph of a linear function.

Objective



A family of functions is a set of functions whose graphs have basic characteristics in common. For example, all linear functions form a family because all of their graphs are the same basic shape.

A parent function is the most basic function in a family. For linear functions, the parent function is f(x) = x.

The graphs of all other linear functions are transformations of the graph of the parent function, f(x) = x. A transformation is a change in position or size of a figure.



There are three types of transformations– translations, rotations, and reflections.

Look at the four functions and their graphs below.



Notice that all of the lines are parallel. The slopes are the same but the y-intercepts are different.



The graphs of g(x) = x + 3, h(x) = x – 2, and k(x) = x – 4, are vertical translations of the graph of the parent function, f(x) = x. A translation is a type of transformation that moves every point the same distance in the same direction. You can think of a translation as a “slide.”



Example 1: Translating Linear Functions

Graph f(x) = 2x and g(x) = 2x – 6. Then describe the transformation from the graph of f(x) to the graph of g(x).

The graph of g(x) = 2x – 6 is the result of translating the graph of f(x) = 2x 6 units down.

f(x) = 2x

g(x) = 2x – 6



Check It Out! Example 1Graph f(x) = x + 4 and g(x) = x – 2. Then describe the transformation from the graph of f(x) to the graph of g(x).

The graph of g(x) = x – 2 is the result of translating the graph of f(x) = x + 4 6 units down.

f(x) = x + 4

g(x) = x – 2



The graphs of g(x) = 3x, h(x) = 5x, and k(x) = are rotations of the graph f(x) = x. A rotation is a transformation about a point. You can think of a rotation as a “turn.” The y-intercepts are the same, but the slopes are different.



Example 2: Rotating Linear FunctionsGraph f(x) = x and g(x) = 5x. Then describe the transformation from the graph of f(x) to the graph of g(x).

The graph of g(x) = 5x is the result of rotating the graph of f(x) = x about (0, 0). The graph of g(x) is steeper than the graph of f(x).

g(x) = 5xf(x) = x




Graph f(x) = 3x – 1 and g(x) = x – 1. Then describe the transformation from the graph of f(x) to the graph of g(x).

The graph of g(x) is the result of rotating the graph of f(x) about (0, –1). The graph of g(x) is less steep than the graph of f(x).

f(x) = 3x – 1

g(x) = x – 1



The diagram shows the reflection of the graph of f(x) = 2x across the y-axis, producing the graph of g(x) = –2x. A reflection is a transformation across a line that produces a mirror image. You can think of a reflection as a “flip” over a line.



Example 3: Reflecting Linear FunctionsGraph f(x) = 2x + 2. Then reflect the graph of f(x) across the y-axis. Write a function g(x) to describe the new graph.

f(x) = 2x + 2

To find g(x), multiply the value of m by –1.In f(x) = 2x + 2, m = 2. 2(–1) = –2g(x) = –2x + 2

f(x)

g(x)

This is the value of m for g(x).



Example 5: Business ApplicationA florist charges $25 for a vase plus $4.50 for each flower. The total charge for the vase and flowers is given by the function f(x) = 4.50x + 25. How will the graph change if the vase’s cost is raised to $35? if the charge per flower is lowered to $3.00?

f(x) = 4.50x + 25 is graphed in blue.

If the vase’s price is raised to $35, the new function is f(g) = 4.50x + 35. The original graph will be translated 10 units up.

Total Cost



Example 5 ContinuedA florist charges $25 for a vase plus $4.50 for each flower. The total charge for the vase and flowers is given by the function f(x) = 4.50x + 25. How will the graph change if the vase’s cost is raised to $35? If the charge per flower is lowered to $3.00?

If the charge per flower is lowered to $3.00. The new function is h(x) = 3.00x + 25. The original graph will be rotated clockwise about (0, 25) and become less steep.

Total Cost



5 Minute Warm-UpDirections: Write an equation in slope intercept form of the line that passes through the points.1. (5, 32) and (7, 16)

Directions: Which of the lines are perpendicular?2. line p: y = 1 x + 2 line q: y = 5x + ½ line r: y = -5x +3 5

Directions: Find the distance and midpoint of the twopoints.3. (-3, 8) and (-10, 5)



Lesson Quiz: Part I

Describe the transformation from the graph of f(x) to the graph of g(x).

1. f(x) = 4x, g(x) = x

2.

3.

4.

rotated about (0, 0) (less steep)

translated 7 units up

reflected across the y-axis

f(x) = x – 1, g(x) = x + 6

f(x) = x, g(x) = 2x

rotated about (0, 0) (steeper)f(x) = 5x, g(x) = –5x



Lesson Quiz: Part II

5. f(x) = x, g(x) = x – 4

6.

translated 4 units down

f(x) = –3x, g(x) = –x + 1rotated about (0, 0) (less steep), translated 1 unit up

7. A cashier gets a $50 bonus for working on a holiday plus $9/h. The total holiday salary is given by the function f(x) = 9x + 50. How will the graph change if the bonus is raised to $75? if the hourly rate is raised to $12/h? translate 25 units up; rotated about (0, 50) (steeper)

Holt Algebra 1 5-1 Identifying Linear Functions Warm Up 1.Solve 2x – 3y = 12 for y. 3.The ratio of...

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