Chapter 2 Functions, Equations, and Graphs
Transcript of Chapter 2 Functions, Equations, and Graphs
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Unit Essential Questions:• Does it matter which form of a
linear equation that you use?• How do you use
transformations to help graph absolute value functions?
• How can you model data with linear equations?
Chapter 2Functions, Equations, and Graphs
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Students will be able to graph relations
Students will be able to identify functions
Section 2.1 Part 1Relations and Functions
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Warm UpGraph each ordered pair on the coordinate
plane.
1) (-4, -8)
2) (3, 6)
3) (0, 0)
4) (-1, 3)
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Key ConceptsRelation - a set of pairs of input and output values.
Four Ways to Represent Relations
Ordered Pairs Mapping Diagram
Table of Values
Graph
(-3, 4)
(3, -1)
(4, -1)
(4, 3)
-3 -13 34 4
x y
-3 4
3 -1
4 -1
4 3
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Example 1When skydivers jump out of an airplane, they experience free fall. At 0 seconds, they are at 10,000ft, 8 seconds, they are at 8976ft, 12 seconds, they are at 7696ft, and 16 seconds, they are at 5904ft. How can you represent this relation in four different ways?
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Example 1 (Continued)Four Ways to Represent the Relation
Ordered Pairs Mapping Diagram
Table of Values
Graph
(0, 10000)
(8, 8976)
(12, 9696)
(16, 5904)
0 100008 897612 969616 5904
Time Height
0 10000
8 8976
12 9696
16 5904
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Key ConceptsDomain - the set of all inputs (x-coordinates)
Range - the set of all outputs (y-coordinates)
Function - a relation in which each element in the domain corresponds to exactly one element of the range.
Vertical Line Test - if any vertical line passes through more than one point on the graph of a relation, then it is not a function.
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Example 2Determine whether each relation is a function.State the domain and range.a) {(0, 1), (1, 0), (2, 1), (3, 1), (4, 2)}
Domain: {0, 1, 2, 3, 4} Yes, it is a function.
Range: {0, 1, 2}
b) {(1, 4), (3, 2), (5, 2), (1, -8), (6, 7)}Domain: {1, 3, 5, 6} No, it is not a function.Range: {-8, 2, 4, 7}
c) {(1, 3), (2, 3), (3, 3), ( 4, 3), ( 5, 3)}Domain: {1, 2, 3, 4, 5} Yes, it is a function.Range: {3}
d) {(4, 9), (4, 3), (4, 0), (4, 4), (4, 1)}Domain: {4} No, it is not a function.Range: {0, 1, 3, 4, 9}
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Example 3
Use the vertical line test. Which graphs represent a function?
Not a Function Function Not a Function
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Students will be able to write and evaluate functions
Section 2.1 Part 2Relations and Functions
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Warm Up1. Can you have a relation that is not a
function?yes, a relation is any set of pairs of input and output values.
2. Can you have a function that is not a relation?
no, a function is a relation.
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Key Concepts
Function Rule - an equation that represents an output value in terms of an input value. You can write the function rule in function notation.
Independent Variable - x, represents the input value.
Dependent Variable - y, represents the output value. (Call dependent because it depends on the input value)
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Example 1Evaluate each function for the given value of x, and write the input x and output f (x) in an ordered pair.
f (x) = -2x + 11 for x = 5, -3, and 0
f(5) = -2(5) + 11f(5) = 1
f(-3) = -2(-3) + 11f(-3) = 17
f(0) = -2(0) + 11f(0) = 11
(5, 1) (0, 11)(-3, 17)
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Example 2
Write a function rule to model the cost per month of a long-distance cell phone calling plan. Then evaluate the function for given number of minutes.
Monthly service fee: $4.99
Rate per minute: $.10
Minutes used: 250 minutes
C (m) = 4.99+ 0.10m C (250) = $29.99
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Students will be able to write and interpret direct variation equations
Section 2.2Direct Variation
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Warm UpSolve each equation for y.
1) 12y = 3x 2) - 10y = 5x 3)
3
4y = 15x
y =
1
4x y = -
1
2x y = 20x
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Key Concepts
Direct Variation - a linear function defined by an equation of the form y=kx, where k ≠ 0.
Constant of Variation - k, where k = y/x.
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Example 1For each function, determine whether yvaries directly with x. If so, find the constant of variation and write the equation.
x –1 2 5y 3 –6 15
a) x 7 9 –4y 14 18 –8
b)
3
-1 = -3
= 3
15
5
-6
2 = -3
Does not vary directly
14
7 = 2
= 2
-8
-4
18
9 = 2
k = 2y = 2x
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Example 2For each function, tell whether y varies directly
with x. If so, find the constant of variation.
a. 3y = 7x + 7 b. 5x = –2y
y =
7
3x +
7
3 Isolate y.
You cannot write the
equation in the form
y = kx.
y does not vary directly
with x.
y = -
5
2x Isolate y.
You can write the
equation in the form
y = kx.
y does vary directly
with x. k = -5
2.
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Example 3Suppose y varies directly with x, and y = 15 when
x = 27. Write the function that models the variation.
Find y when x = 18.
y = kx
15 = k(27)
k =
5
9
y =
5
9x
y =5
9(18)
y = 10
Find k. Write the function that models the variation.
Find y when x = 18.
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Students will be able to graph linear equations
Students will be able to write equations of line
Section 2.3Linear Functions and Slope-Intercept Form
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Warm UpEvaluate each expression for x = –2, and 0.
1. f(x) = 2x + 7 2. f(x) = 3x – 2
f (-2) = 2(-2) + 7
f (-2) = 3
(-2,3)
f (0) = 2(0) + 7
f (0) = 7
(0,7)
f (-2) = 3(-2) -2
f (-2) = -8
(-2,-8)
f (0) = 3(0) -2
f (0) = -2
(0,-2)
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Key Concepts
Slope - the rate of change
Slope = vertical change (rise)
horizontal change (run)=
y2
- y1
x2
- x1
Positive Negative Zero Undefined
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Key Concepts
Linear Function - a function whose graph is a line
Linear Equation - represents a linear function where a solution is any ordered pair (x,y) that makes the equation true.
x-intercept - the point in which a line crosses the y-axis
y-intercept - the point in which a line crosses the x-axis
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Key Concepts
Slope-Intercept Form
y = mx + bm = slope
(0, b) = y-intercept
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Example 1Find the slope of the line through the points:
a) (3, 0) and (5, 8)
b) (1, –4) and (2, –5)
c) (–2, 7) and (8, –6)
8 - 0
5 - 3 =
8
2
-5 - (-4)
2 - 1 = -
1
1
-6 - 7
8 - (-2) = -
13
10
= -1
= 4
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Example 2What is an equation of the line that has a slope of 1/5 and the y-intercept is (0, -3)?
y =
1
5x - 3
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Example 3Write the equation in slope-intercept form and then find the slope and y-intercept of
–7x + 2y = 8.
2y = 7x + 8
y =
7
2x + 4
Slope = 7/2, y-intercept = (0, 4)
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Example 4Graph the equation y = 2x + 1.
Draw a point on the y-intercept (0, 1)
Draw a point up 2 units and right 1 unit from the y-intercept
Draw a line through the two points
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Students will be able to write equations of lines
Section 2.4 Part 1More about Linear Equations
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Warm UpEvaluate for x = 0.
1. 5x + 2
2. (x - 4) + 12
3. 13 - 6.5x
5(0) +2 = 2
(0 - 4) + 12 = 8
13- 6.5(0) = 13
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Key Concepts
Point-Slope Form
y-y1 = m(x-x1)
Use this form when you are given a point
(x1, y1) and the slope (m).
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Example 1A line passes through (-4,1) with slope 2/5. What is the equation of the line in point-slope form?
(y - 1) =
2
5(x + 4)
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Example 2Write in point-slope form an equation of the
line through (4, –3) and (5, –1).
-1 - (-3)
5 - 4
Find the slope
2
Use a point and the slope to write in point-slope form
(y + 3) = 2(x - 4)
or
(y + 1) = 2(x - 5)
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Key Concepts
Standard Form of a Linear Equation
Ax + By = C where A, B, and C are real numbers, and A and B are
not both zero.
Steps to writing an equation in standard form:1) Multiply each term to clear fractions or decimals2) Isolate the variables on the left side of the
equation
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Example 3Write each equation in standard form. Use integer coefficients.
a. y = 3/4x - 5 b. y = -4.2x - 5.5
Multiply by 4 4y = 3x - 20
-3x + 4y = -20Subtract3x
Multiply by -1 3x - 4y = 20
Multiply by 10 10y = -42x - 55
42x + 10y = -55Add 42x
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Students will be able to write and graph the equation of a line
Students will be able to write equations of parallel and perpendicular lines
Section 2.4 Part 2More about Linear Equations
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Warm UpFind the slope through the following points.
1. (2, 4) and (-1, 5) 2. (-7, -8) and (-2, -8)
5 - 4
-1 - 2
-
1
3
-8 - (-8)
-2- (-7)
0
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Key ConceptsParallel Lines - the slopes of parallel lines are equal.
m1 = m2
Perpendicular Lines - the slopes of perpendicular lines are negative reciprocals of each other.
m1 = - 1/m2
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Example 1What are the intercepts of 3x + 5y = 15? Graph the equation.
3x + 5(0) = 15Set y = 0to find the x-intercept 3x = 15
x = 5
(5, 0)
Set x = 0to find the y-intercept
3(0) + 5y = 15
5y = 15
y = 3
(0,3)
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Example 2Write in slope-intercept form an equation of the line through (1, –3) and parallel to y = 6x - 2. Graph to check your answer.
Slope for the equation y = 6x – 2 is 6. Therefore, the slope of our line must be 6.
Put in point-slope form (y + 3) = 6(x - 1)
Distribute 6 y + 3 = 6x - 6
Subtract 3 y = 6x - 9
y = 6x - 9
y = 6x -2
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Example 3Write in slope-intercept form an equation of the line through (8, 5) and perpendicular to y = -4x + 6. Graph to check your answer.
Slope for the equation y = -4x + 6 is -4. Therefore, the slope of our line must be 1/4.
Put in point-slope form
(y - 5) =
1
4(x - 8)
Distribute 1/4
y - 5 =
1
4x - 2
Add 5
y =
1
4x + 3
y =
1
4x + 3
y = -4x + 6
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Students will be able to analyze transformations of functions
Section 2.6 Part 1Families of Functions
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Warm UpGraph each pair of functions on the same
coordinate plane.
1) y = -x 2) y = x + 1 3) y = -1/4x
y = x y = 2x - 1 y = -1/4x + 2
y = x
y = -x
y = 2x - 1 y = x + 1
y = -
1
4x + 2
y = -
1
4x
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Key Concepts
Parent Function - the simplest form in a set of functions that form a family
Translation - operation that shifts a graph horizontally, vertically, or both.
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Key Concepts
Vertical Translation - moves the graph up or down, represented by k.
y = f(x) ± kAddition moves it up, subtraction moves it down.
(Same Sign)
Horizontal Translation - moves the graph right or left, represented by h.
y = f(x ± h)Addition moves it left, subtraction moves it right.
(Opposite Sign)
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Example 1Describe how the functions y = 2x and y = 2x + 3 are related. How are their graphs related?
y = 2x + 3 is a vertical shift up 3 units of the parent function y = 2x.
Graphs are both linear and parallel.
y = 2x + 3
y = 2x
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Example 2Make a table of values for f(x) = x2 and then for
g(x) = (x – 5)2. Describe the transformation.
x f(x) = x2 g(x) = (x – 5)2
-2 4 49
-1 1 36
0 0 25
1 1 16
2 4 9
3 9 4
4 16 1
5 25 0
g(x) is shifted horizontally 5 units to the right from f(x).
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Example 3a. Write an equation to translate the graph of y = 4x,
5 units down.
b. Write an equation to translate the graph of y = 6x, 3 units to the right.
y = 4x - 5
y = 6(x - 3)
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Example 4Write an equation for each translation of y = x2.
a) 3 units up, 7 units right
b) 5 units down, 1 unit left
y = (x - 7)2 + 3
y = (x + 1)2 - 5
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Students will be able to analyze transformations of functions
Section 2.6 Part 2Families of Functions
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Warm UpGraph each pair of functions on the same coordinate plane.
1) y = x, y = x + 4 2) y = –2x, y = –2x – 3 3) y = x, y = x – 2
y = x + 4
y = x
y = -2x - 3
y = -2x y = x -2
y = x
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Key Concepts
Reflection - flips the graph of a function across a line such as the x- or y-axis.
*For the function f(x), the reflection in the y-axis is f(-x).
*For the function f(x), the reflection in the x-axis is –f(x)
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Example 1Write the function rule for each function reflected in the given axis.
a) f(x) = 2x – 1 x-axis:
y-axis:
b) f(x) = x + 3 x-axis:
y-axis:
y = -2x + 1
y = -2x - 1
y = -x - 3
y = -x + 3
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Key Concepts
Vertical Stretch - multiplies all y-values of a function by the same factor greater than 1.
Vertical Compression - reduces all y-values of a function by the same factor between 0 and 1.
In other words,
let f (x) be the parent function,
• if a > 1 then y = af(x) is a vertical stretch
• if 0 < a < 1 then y = af(x) is a vertical compression
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Example 2
a) y = x 2
b) y = 3x 2
c) y =1
3x 2
Graph each of the following on the same coordinate plane and create a table of values for each. Explain the transformations from the first equation.
y = 3x 2
y =
1
3x 2 y = x 2
Stretch by 3.
Compress by 1/3.
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Example 3The graph of g(x) is the graph of f(x) = 6x compressed vertically by the factor 1/2 and then reflected in the y-axis. What is the function rule for g(x)?
g(x) = -3x
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Example 4What transformations change the graph of f(x) to
the graph of g(x)?
f(x) = 2x2 g(x) = 6x2 – 1
Stretch by 3 unitsVertical shift down 1 unit
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Concept SummaryTransformations of f(x)
Vertical Translation
(k units)
Up: y = f(x) + k
Down: y = f(x) - k
Horizontal Translation
(h units)
Right: y = f(x - h)
Left: y = f(x + h)
Vertical Stretch/Compression
Stretch (a>1): y = af(x)
Compression (0<a<1): y = af(x)
Reflection
x-axis: y = -f(x)
y-axis: y = f(-x)
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Students will be able to graph absolute value functions
Section 2.7Absolute Value Functions and Graphs
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Warm UpSolve each absolute value equation.
1) x - 3 + 2 = 7 2)
1
35x - 3 = 6
x - 3 = 5
x = 8
x - 3 = -5
x = -2
|x - 3| = 5
5x - 3 = 18
x = 21/5
5x - 3 = -18
x = -3
|5x - 3| = 18
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Key ConceptsGraph of Absolute Value Function
f (x) = x
x y
-2 2
-1 1
0 0
1 1
2 2
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Key Concepts
Axis of Symmetry - the line that divides a figure into two parts that are mirror images
Vertex - a point where the function reaches a maximum or minimum value
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Example 1What is the graph of the absolute value function
y = x - 4
y = x - 4
y = x
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Key ConceptsTransformations of y = |x|
Vertical Translation
(k units)
Up: y = |x| + k
Down: y = |x| - k
Horizontal Translation
(h units)
Right: y = |x – h|
Left: y = |x + h|
Vertical Stretch/Compression
Stretch (a>1): y = a|x|
Compression (0<a<1): y = a|x|---
Reflection
x-axis: y = -|x| =====
-----
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Example 2Graph each absolute value function.
a) y = x + 2 + 3 b) y =
1
2x
y = x
y = x
y = x + 2 + 3
y =
1
2x
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Key Concepts
General Form of the Absolute Value Function
Stretch/Compression Factor is |a| ---Vertex is (h, k)
Axis of Symmetry is x = h
y = a x - h + k
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Example 3Without graphing, identify the vertex, axis of symmetry, and transformations of
f(x) = -3|x – 1| + 4 from the parent function
f(x) = |x|.
Vertex: (1, 4)Axis of Symmetry: x = 1Stretch by 3 unitsReflect over the x-axisHorizontal shift right 1 unitVertical shift up 4 units
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Example 4Write an absolute value equation for the
given graph.
y =
1
4x - 2 + 1
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Students will be able to graph two-variable inequalities
Section 2.8Two Variable Inequalities
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Warm UpSolve each inequality. Graph the solution on a number line.
1. 12p £ 15 2. 4 +t > 17 3. 5 -2t ³ 11
p ≤ 5/4 t > 13 -2t ≥ 6t ≤ -3
5/4 -313
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Key ConceptsLinear Inequality -an inequality in two variables
whose graph is a region of the coordinate plane that is bounded by a line.
Steps to graph two-variable inequality1) Graph the boundary line (graph as if it was an
equation) • If y< or y>, then the boundary is dashed. If y≥
or y≤, then the boundary is solid.2) Shade the solutions
• For y< or y≤, shade below the boundary. For y> or y≥, shade above the boundary.
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Example 1
Graph y £ 3x - 1
Graph y = 3x – 1 for theboundary line
The boundary line is solid because the inequalityis ≤.
Shade below the boundary line.
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Example 2
Graph y >
1
2x + 3
Graph y = 1/2x + 3 for theboundary line
The boundary line is dashed because the inequalityis >.
Shade above the boundary line.
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Example 2
Graph
1 + y < x + 2
Graph y = |x + 2| - 1 for theboundary line
The boundary line is dashed because the inequalityis <.
Shade below the boundary line.