Section 8.2 Linear Functions. 8.2 Lecture Guide: Linear Functions.

Section 8.2Linear Functions

8.2 Lecture Guide: Linear Functions

Key Characteristics of a Linear Function

1. The graph of a linear function is a ____________ line.

2. The equation is first degree and can be written in the slope-intercept form with slope ______ and y-intercept ______.

3. Linear functions have a constant rate of change. The slope is the same between any two ____________ on the line.

4. The domain of all linear functions is ______. The range of all linear functions is also unless the function is a constant function, whose graph is a horizontal line.

f x mx b

Algebraically Numerical Example

Graphical Example

A function of the form f x mx b

is called a linear function.

1 7

0 5

1 3

2 1

3 1

4 3

5 5

x y f x

-6 6

-6

6y

x

Linear Function

2 5f x x

Objective 1: Determine the slope of a line.

From Section 3.1, we define slope of a line through two points as the ratio of the change in y to the change in x. This rise over the run can be expressed algebraically by or A line defined by the slope-intercept

Form has a slope m.

2 1

2 1

y ym

x x

.y

mx

f x mx b

Determine the slope of each line.

5. 6. 7. 37

5f x x 3f x 4 5f x x

8. Calculate the slope of the line in the graph.


-5 5

-5

5y

x

-5 5

-5

5y

x

12. Calculate the slope of the line containing the points in the table.

0 2

5 5

10 8

15 11

20 14

25 17

30 20

x y

3 4

0 2

3 0

6 2

9 4

12 6

15 8

x y

13. Calculate the slope of the line containing the points in the table.

Interpreting Slopes

Positive Slope Zero Slope Negative Slope

Algebraically Algebraically Algebraically

In m will be positive.

Example:

In m will be 0 and

Example:

In m will be negative.

Example:

,f x mx b

11

2f x x 4f x

,f x mx b ,f x mx b

.f x b

2 1f x x

Interpreting Slopes


Graphically Graphically Graphically

The line will slope ____________ to the right.

Example:

This is a ________ line that does not slope upward or downward.

Example:

The line slopes ____________ to the right.

Example:

Interpreting Slopes


Numerically Numerically Numerically

The y-values will ____________ as the x-values increase.

The y-values will _______________ as the x-values change.

The y-values will ____________ as the x-values increase.

4 3

2 2

0 1

2 0

4 1

x y

2 4

1 4

0 4

1 4

2 4

x y

2 5

1 3

0 1

1 1

2 3

x y

The equation of a vertical line is of the form This equation does not represent a function --- it fails the vertical line test. The slope of a vertical line is ____________.

.x k

Objective 2: Sketch the graph of a linear function.

Use the slope and y-intercept to graph each line.

14. 35

4f x x

Slope: ______

y-intercept: ______

Graph:

-6

6

-6 6

y

x

Use the slope and y-intercept to graph each line.

15.

Slope: ______

y-intercept: ______

Graph:

-6

6

-6 6

y

x

13

2f x x

Objective 3: Write the equation of the line through given points.

Determine the equation of the line in the form that passes through each pair of points.

f x mx b

16. and 3,4 0,7

Determine the equation of the line in the form that passes through each pair of points.

f x mx b

17. and 1, 2 4, 6

All the points listed in the table lie on the same line. Use the table to determine the equation of the line in the form .f x mx b

2 5

0 2

2 1

4 4

6 7

x y

18.

All the points listed in the table lie on the same line. Use the table to determine the equation of the line in the form .f x mx b

19. 3 2

1 3

5 8

9 13

13 18

x y

Use the information displayed in the graph to determine the equation of the line in the form .f x mx b

20.

0,1

2, 3

Use the information displayed in the graph to determine the equation of the line in the form .f x mx b

21.

2, 3 1, 1

Objective 4. Determine the intercepts of a line.

22. Use the graph to determine the intercepts of thelinear function.

Objective 4. Determine the intercepts of a line.

23. Use the table to determine the interceptsof the linear function.

3 5

2 0

1 5

0 10

1 15

2 20

3 25

x y

24. Determine the intercepts of the linear function. Then use the intercepts to sketch a graph of the function.

26

3f x x

Objective 5: Determine the x-values for which a linear function is positive and the x-values for which a linearfunction is negative.

Classifying a Function as Positive or Negative

Verbally Graphically Numerically Algebraically

The function

is positive

at

is

__________

the x-axis.

has a

positive

y-value.

The function

is negative

at

is

__________

the x-axis.

has a

negative

y-value.

,x y ,x y

0f x ,x y

, .x y

,x y

0f x

, .x y

Determine the x-values for which each linear function is positive and the x-values for which each function is negative.

25.


26.

3 5

2 0

1 5

0 10

1 15

2 20

3 25

x y

27. 12

3f x x


28. Determine the profit and loss intervals for the profit function graphed below. The x-variable represents the

number of units of production and y-variable represents the profit generated by the sale of this production.

-200

-100

0

100

200

300

0 50 100 150

Profit interval:

Loss interval:Units

Profit

Write in slope-intercept form the equation of a line passing

through and perpendicular to

29.

3,1 35.

4y x

30. Use the function to determine the missing input and output values.

4 8f x x

(a) (b) 4 ______f 4; ______f x x

31.Temperature Beginning at 6:00 am, when the temperature was Fahrenheit, the temperature increased by per hour over the next 6 hours.

388

(a) Write a linear function T so that T(x) gives the temperature in degrees

(b) Determine the temperature at noon.

(c) Use this function to determine how many hours until the temperature reaches .74


388

d) Use this function to complete the table.

0

1

2

3

4

5

6

x T x


388

Section 8.2 Linear Functions. 8.2 Lecture Guide: Linear Functions.

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Transcript of Section 8.2 Linear Functions. 8.2 Lecture Guide: Linear Functions.