9.11.1 Maximum and Minimum Value of a Function

Algebra 19.11.1 Maximum and Minimum Value of a Function

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Warm Up

Examine the graph that represents the function, $f$ , defined by $f (x) = (x - 4)^{2} + 1$ .

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

a. Enter the value of $f$ when $x$ is 1

b. Enter the value of $f (3)$ .

$f (3)$

c. Enter the value of $f (10)$ .

$f (10)$

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

a. Less than 1?

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

b. Greater than 10,000?

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

Examine the graph that represents the function, $g$ , defined by $g (x) = - (x - 12)^{2} + 7$ .

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

a. Enter the value of $g$ when $x$ is 9

-2

b. Enter the value of $g (13)$ .

$g (13)$

c. Enter the value of $g (2)$ .

$g (2)$

-93

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

a. Greater than 7?

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

b. Less than -10,000?

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

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
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- Section URL: https://openstax.org/books/algebra-1/pages/9-11-1-maximum-and-minimum-value-of-a-function

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