Section 8.5 Analyzing Graphs

8.5 Lecture Guide: Analyzing Graphs

Objective 1: Identify a function as an increasing or decreasing function.

Graphical Example Verbally

A function is increasingover its entire domain ifthe graph rises as itmoves from left toright.

For all x-values, as thex-values increase, they-values also increase.

Increasing Function

y

x

y f x

Increasing and Decreasing Functions

Graphical Example Verbally

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.

Decreasing Function

Increasing and Decreasing Functions

y

x y f x

-6

6

-6 6

x

y

1. 2. 3.

-6 6

-6

6y

x

-6 6

-6

6y

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.

Classification of Slopes Positive Slope: An Increasing Linear Function

Negative Slope:A Decreasing Linear Function

Zero Slope: A Constant Linear Function

For an increasing function, a positive value of produces a ____________ value of .

For a decreasing function, a positive value of produces a ____________ value of .

For a constant function, any change in x produces ______ change in y.

x

y

x

y

4. 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.

5. 5 7f x 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.

6. 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.

7. 3 12f x x

8. The graph gives the altitude y in thousands of feet in terms of the time x in minutes since the start of a flight in a small plane.

(a) During what time interval was the altitude increasing?

(b) During what time interval was the altitude constant?

(c) During what time interval was the altitude decreasing?

10 20 30 40 50 60 700

1

2

3

4

0x

y

Minutes

Altitude in thousands

of feet

Objective 2: Analyze the graphs of linear, absolute value, and quadratic functions.

The following box contains a check list of key information that we are currently able to obtain about the graphs of linear, absolute value, and quadratic functions. We will use this list as a guide for analyzing each of these functions and their graphs.

1. Identify the type of function.a. For linear functions, determine the slope.b. For absolute value and quadratic functions, determine

the vertex and whether the graph opens upward or downward.

2. Determine the y-intercept and the x-intercept(s).

3. Determine the domain and range.

4. Determine where the function is positive and where the function is negative.

5. Determine where the function is increasing and where the function is decreasing.

Analyzing the Graphs of Linear, Absolute Value, and Quadratic Functions

(a) Determine the slope of this line.

(b) Determine the x-intercept.

(c) Determine the y-intercept.

(d) Determine the domain of this function.

-5 5

-5

5y

x

9. Use the graph of the linear function to determine the missing information.

(e) Determine the range of this function.

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

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

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

(i) Determine the x-values for which the function is increasing.

-5 5

-5

5y

x

9. Use the graph of the linear function to determine the missing information.

10. Use the linear function to determine the missing information.

25 300f x x

(a) Determine the slope of this line.

(b) Determine the x-intercept.

(c) Determine the y-intercept.

(d) Determine the domain of this function.

10. Use the linear function to determine the missing information.

25 300f x x

(e) Determine the range of this function.

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

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

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

(i) Determine the x-values for which the function is increasing.

11. Use the graph of the absolute value function to determine the missing information.

(a) Determine the vertex of this graph.

(b) Does the graph open upward or downward?

(c) Determine the x-intercept(s).

(d) Determine the y-intercept.

(e) Determine the domain of this function.

11. Use the graph of the absolute value function to determine the missing information.

(f) Determine the range of this function.

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

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

(i) Determine the x-values for which the function is decreasing.

(j) Determine the x-values for which the function is increasing.

(a) Determine the vertex of this graph.

(b) Does the graph open upward or downward?

(c) Determine the x-intercept(s).

(d) Determine the y-intercept.

(e) Determine the domain of this function.

12. Use the absolute value function to determine the missing information.

20 12f x x

12. Use the absolute value function to determine the missing information.

(f) Determine the range of this function.

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

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

(i) Determine the x-values for which the function is decreasing.

(j) Determine the x-values for which the function is increasing.

20 12f x x

13. Use the graph of the quadratic function to determine the missing information.

(a) Determine the vertex of this graph.

(b) Does the graph open upward or downward?

(c) Determine the x-intercept(s).

(d) Determine the y-intercept.

(e) Determine the domain of this function.

(f) Determine the range of this function.

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

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

(i) Determine the x-values for which the function is decreasing.

(j) Determine the x-values for which the function is increasing.

13. Use the graph of the quadratic function to determine the missing information.

14. Use the quadratic function to determine the missing information.

2 7 30f x x x

(a) Determine the vertex of this graph.

(b) Does the graph open upward or downward?

(c) Determine the x-intercept(s).

14. Use the quadratic function to determine the missing information.

2 7 30f x x x

(d) Determine the y-intercept.

(e) Determine the domain of this function.

(f) Determine the range of this function.

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

14. Use the quadratic function to determine the missing information.

2 7 30f x x x

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

(i) Determine the x-values for which the function is decreasing.

(j) Determine the x-values for which the function is increasing.

15. The height in feet of a ball thrown off the roof of a building is given by where t represents the time in seconds since the ball was thrown. This function is also represented in the graph and table below. Answer each question below algebraically. Use the graph and the table to verify your results.

216 60 75h t t t

0

25

50

75

100

125

150

0 1 2 3 4 5

y

x

Time (sec)

Height (ft)

0.0 750.5 1011.0 1191.5 1292.0 1312.5 1253.0 1113.5 894.0 594.5 21

Time (sec)

Hei

gh

t (f

t)

15. The height in feet of a ball thrown off the roof of a building is given by where t represents the time in seconds since the ball was thrown. This function is also represented in the graph and table below. Answer each question below algebraically. Use the graph and the table to verify your results.

216 60 75h t t t

Time (sec)

Height (ft)

0.0 750.5 1011.0 1191.5 1292.0 1312.5 1253.0 1113.5 894.0 594.5 21

(a) Determine the y-intercept and interpret its meaning.

(b) Determine the x-intercept and interpret its meaning.

15. The height in feet of a ball thrown off the roof of a building is given by where t represents the time in seconds since the ball was thrown. This function is also represented in the graph and table below. Answer each question below algebraically. Use the graph and the table to verify your results.

216 60 75h t t t

Time (sec)

Height (ft)

0.0 750.5 1011.0 1191.5 1292.0 1312.5 1253.0 1113.5 894.0 594.5 21

(c) Determine the vertex and interpret the meaning of each coordinate.

(d) Determine the t-values for which the function is increasing.

 

15. The height in feet of a ball thrown off the roof of a building is given by where t represents the time in seconds since the ball was thrown. This function is also represented in the graph and table below. Answer each question below algebraically. Use the graph and the table to verify your results.

216 60 75h t t t

Time (sec)

Height (ft)

0.0 750.5 1011.0 1191.5 1292.0 1312.5 1253.0 1113.5 894.0 594.5 21

(e) Interpret the meaning of the interval in part (d).