Using Quadratic Expressions in Vertex Form to Solve Problems

10 min

Narrative

In this activity, students recall the meaning of maximum or minimum value of a function, which they learned in a previous unit. They also practice interpreting the language related to maximum and minimum values of functions.

Launch

Ask students to describe some situations in which people use the words “minimum” and “maximum.” For example, we might say there is a minimum age for voting or for getting a driver’s license, or that roads and highways have maximum speed limits.

Then, ask students what the words “minimum” and “maximum” mean more generally. We might think of a minimum as the least, the least possible, or the least allowable value, and a maximum as the greatest, the greatest possible, or the greatest allowable value.

Student Task

Here are graphs that represent two functions, ff and gg, defined by these equations:

f(x)=(x−4)2+1f(x)=(x-4)^2+1

g(x)=-(x−12)2+7g(x)=\text-(x-12)^2+7

<p>Function f and g on a grid.</p>
Function f and g on a grid. X axis from 0 to 15, by 1’s. Y axis from 0 to 10, by 2’s. Parabola f opens upward with vertex at 4 comma 1. Additional points on the parabola are 1 comma 10, 2 comma 5, 3 comma 2, 5 comma 2, 6 comma 5 and 7 comma 10. Parabola g opens downward with a vertex at 12 comma 7. Additional points on the parabola are 9 comma negative 1, 10 comma 3, 11 comma 6, 13 comma 6, 14 comma 3, and 15 comma negative 1.

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

    1. the value of ff when xx is 1
    2. f(3)f(3)
    3. f(10)f(10)
  2. Does ff have a maximum, minimum, or neither? If it has a maximum or minimum, what is the greatest or least value f(x)f(x) can have?

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

    1. the value of gg when xx is 9
    2. g(13)g(13)
    3. g(2)g(2)

  4. Does gg have a maximum, minimum, or neither? If it has a maximum or minimum, what is the greatest or least value g(x)g(x) can have?

Sample Response

  1. Function ff:
    1. 10
    2. 2
    3. 37
  2. minimum, with a value of 1
  3. Function gg:
    1. -2
    2. 6
    3. -93
  4. maximum, with a value of 7

Synthesis

Discuss with students:

  • “Why does ff not have a maximum value?” (We can always use larger values of xx in both the positive and negative directions to get greater and greater values of ff.)
  • “Why does gg not have a minimum value?” (We can always find an input that makes the value of gg less and less.)

Emphasize that we can find an input that makes the value of ff as great as we want and that makes gg as small as we want.

Remind students that:

  • A maximum value of a function is a value of a function that is greater than or equal to all the other values. It corresponds to the highest yy-value on the graph of the function. 
  • A minimum value of a function is a value of a function that is less than or equal to all the other values. It corresponds to the lowest yy-value on the graph of the function.
  • For quadratic functions, there is only one maximum or minimum value.
Anticipated Misconceptions

Some students may struggle to relate the yy-coordinates of points on a graph with the outputs of a function. Earlier in the course, students learned that the graph of a function ff is the graph of the equation y=f(x)y=f(x). Consider having students label the coordinates of points on each graph and then complete the statements such as “The point (3,2)(3,2) on the graph means 2=f(3)2=f(3).” Another approach would be to have students organize the points on the graphs into tables with headers xx and f(x)f(x) and xx and g(x)g(x).

Standards
Building On
  • F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • F-IF.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  • HSF-IF.A.2·Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Addressing
  • F-IF.C·Analyze functions using different representations
  • HSF-IF.C·Analyze functions using different representations.
Building Toward
  • A-SSE.3.b·Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • A-SSE.3.b·Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • A-SSE.3.b·Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
  • HSA-SSE.B.3.b·Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

10 min

15 min