Section 7.2:

Linear and Absolute Value Functions

7.2 Lecture Guide: Linear and Absolute Value Functions

Objective: Use the slope-intercept form of a linear equation.

Linear Function

Algebraically

A function of the form f x mx b is called a linear function.

Graphical Example 2 5f x x

Verbally The graph of 2 5y x is a straight line. Each point on this line satisfies this equation.

This line rises 2-units for each 1-unit move to the right.

Linear Function

Algebraically

A function of the form f x mx b is called a linear function.

Numerical Example 2 5f x x

1 7

0 5

1 3

2 1

3 1

4 3

5 5

x y f x

Verbally In the table for 2 5y x , each

1-unit increase in x produces a 2-unit increase in y.

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

1. 35

4f x x

Slope: ______

y-intercept: ______

Graph:

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

2.

Slope: ______

y-intercept: ______

Graph: 13

2f x x

Use the given graph or table to determine the slope of the line, the y-intercept of the graph, and the equation of the line in slope-intercept form.

3. Graph:

Slope: ______

y-intercept: ______

Equation: __________________

Use the given graph or table to determine the slope of the line, the y-intercept of the graph, and the equation of the line in slope-intercept form.

4. Table:

Slope: ______

y-intercept: ______

Equation: __________________

Objective: Identify a function as an increasing or decreasing function.

Increasing and Decreasing Functions

Graphical Example

Verbally

Increasing Function

y

x

y f x

A function is increasing over its entire domain if the graph rises as it moves from left to right.

For all x-values, as the x-values increase, the y-values also increase.

Increasing and Decreasing Functions

Graphical Example

Verbally

Decreasing Function

y

x

y f x

A function is decreasing over its entire domain if the graph drops as it moves from left to right.

For all x-values, as the x-values increase, the y-values decrease.

5.

-5

5

-5 5

y

x

Use the graph of each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing.

Use the graph of each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing.

6.

-5

5

-5 5

y

x

7.

-5

5

-5 5

y

x

Use the graph of each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing.

8. For a linear function in the form f x mx b :

(a) The function is increasing if __________________.

(b) The function is decreasing if __________________.

(c) The function is neither increasing nor decreasing if __________________.

Use the equation defining each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing. Use a graphing calculator only as a check.

9. 5 7f x x

10. 10f x

Use the equation defining each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing. Use a graphing calculator only as a check.

11.

Use the equation defining each function to identify the function as an increasing function, a decreasing function, or a function that is neither increasing nor decreasing. Use a graphing calculator only as a check.

3 12f x x

Objective: Determine where a function is positive or negative.

Objective: Analyze the graph of an absolute value function.

Verbally

Graphically Algebraically

Absolute Value Function

Numerically

f x x

3 3

2 2

1 1

0 0

1 1

2 2

3 3

x f x x

-5

5

-5 5

x

y

This V-shaped graph opens upward with thevertex of the V-shape at 0,0 . The minimumy-value is 0. The domain, the projection ofthis graph onto the x-axis, is , .D The range, the projection of this graph ontothe y-axis, is [0, ).R The function isdecreasing for 0x and increasing for

0.x

Vertex

f x x

Complete the table and graph each absolute value function.

12. Equation: 2f x x

-5

5

-5 5

y

x

3

2

1

0

1

2

3

x f x

(a) Does the graph open up or down?

(b) Determine the vertex.

(c) Determine the maximum or minimum y-value.

(d) Determine the domain.

(e) Determine the range.

Complete the table and graph each absolute value function.

13. Equation: 2f x x

2

1

0

1

2

3

4

x f x

-5

5

-5 5

y

x

(a) Does the graph open up or down?

(b) Determine the vertex.

(c) Determine the maximum or minimum y-value.

(d) Determine the domain.

(e) Determine the range.

Use the graph of each absolute value function to determine the interval of x-values for which the function is increasing and the interval of x-values for which the function is decreasing.

14.

-5

5

-5 5

y

x

Increasing____________

Decreasing____________

-5

5

-5 5

y

x

Use the graph of each absolute value function to determine the interval of x-values for which the function is increasing and the interval of x-values for which the function is decreasing.

15.

Increasing____________

Decreasing____________

Positive and Negative Functions

Positive Function:

A function y f x is positive when the output value f x

is positive.

On the graph of y f x this occurs at points above the

x-axis. The y-values are all positive.

Negative Function:

A function y f x is negative when the output value f x

is negative.

On the graph of y f x this occurs at points below the x-axis.

The y-values are all negative.

-5

5

-5 5

y

x

16.(a) Determine the x-values for which the function is positive.

(b) Determine the x-values for which the function is negative.

-5

5

-5 5

y

x

(a) Determine the x-values for which the function is positive.

17.

(b) Determine the x-values for which the function is negative.

Complete the table, graph the absolute value function and determine the following.

1 2f x x 18.

2

1

0

1

2

3

4

x f x

-5

5

-5 5

y

x

Complete the table, graph the absolute value function and determine the following.

1 2f x x 18.

(a) Vertex

(b) The maximum or minimum y-value

(c) Domain

(d) Range

Complete the table, graph the absolute value function and determine the following.

1 2f x x 18.

(e) The x-values for which the function is positive

(f) The x-values for which the function is negative

Complete the table, graph the absolute value function and determine the following.

1 2f x x 18.

(g) The x-values for which the function is increasing

(h) The x-values for which the function is decreasing

Absolute Value Expression

Verbally

Graphically

Equivalent Expression

Solving Absolute Value Equations and Inequalities

For any real numbers x and a and positive real number d:

x a d

x is either d units left or right of a.

orx a d

x a d

a − d a a + d

Absolute Value Expression

Verbally

Graphically

Equivalent Expression

Solving Absolute Value Equations and Inequalities

For any real numbers x and a and positive real number d:

x a d

x is less than d units from a.

d x a d

( )a − d a a + d

Absolute Value Expression

Verbally

Graphically

Equivalent Expression

Solving Absolute Value Equations and Inequalities

For any real numbers x and a and positive real number d:

x a d

x is more than d units from a.

orx a d

x a d

a − d a a + d()

Similar statements can also be made about the order relations less than or equal to and greater than or equal to

. Expressions with d negative are examined in the groupexercises at the end of this section.

Solve each equation and inequality.

19. 2 1 5x

Solve each equation and inequality.

20. 76x

Solve each equation and inequality.

21. 4 10x

Solve each equation and inequality.

22. 92 3x

23. 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

Units

Profit interval:

Loss interval: