Maximum and Minimum Value of a Function

9.11.1 Maximum and Minimum Value of a Function

9.11.1 • Maximum and Minimum Value of a Function

Warm Up

Examine the graph that represents the function, ff, defined by f(x)=(x−4)2+1f(x)=(x−4)2+1.

Graph of a parabola on a coordinate plane. The parabola has been labeled as function f and the plots for 7 points on the graph have been emphasized but not labeled. The x-axis has a scale of 1 extending from 0 to 18. The y-axis extends from negative 4 to 12 with a scale of 1.

Use this graph to answer questions 1 – 2.

1. f(1)f(1) can be expressed in words as "the value of ff when xx is 1." Find or compute:

a. The value of ff when xx is 1.

10

b. Find the value of f(3)f(3).

2

c. Find the value of f(10)f(10).

37

2. Can you find an xx-value that would make f(x)f(x):

a. Less than 1?

No, from the graph, it seems that 1 is the least value of ff, and that occurs when x=4x=4. All other xx-values produce outputs that are greater than 1.

b. Greater than 10,000?

Yes, there are lots of xx-values that produce an output greater than 10,000. For example, both -100 and 110 would make f(x)f(x) greater than 10,000.

Examine the graph that represents the function, gg, defined by g(x)=−(x−12)2+7g(x)=−(x−12)2+7.

Graph of a parabola on a coordinate plane. The parabola has been labeled as function g and the plots for 7 points on the graph have been emphasized but not labeled. The x-axis has a scale of 1 extending from 0 to 18. The y-axis extends from negative 4 to 12 with a scale of 1.

Use this graph to answer questions 3 and 4.

3. g(9)g(9) can be expressed in words as "the value of gg when xx is 9." Find or compute:

a. Find the value of gg when xx is 9

-2

b. Find the value of g(13)g(13).

6

c. Find the value of g(2)g(2).

-93

4. Can you find an xx-value that would make g(x)g(x):

a. Greater than 7?

No, it seems that 7 is the greatest value of gg, and that occurs when x=12x=12. All other xx-values produce outputs that are less than 7.

b. Less than -10,000?

Yes, there are lots of xx-values that produce an output less than -10,000. For example, both -200 and 200 would make g(x)g(x) greater than -10,000.