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Algebra 1

9.11.2 Does the Vertex Represent the Minimum or Maximum Value?

Algebra 19.11.2 Does the Vertex Represent the Minimum or Maximum Value?

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Activity

1. The graph that represents p(x)=(x8)2+1p(x)=(x8)2+1 has its vertex (of a graph) at (8,1)(8,1). Here is one way to show, without graphing, that (8,1)(8,1) corresponds to the minimum value of pp.

  • When x=8x=8, the value of (x8)2(x8)2 is 0 because (88)2=02=0(88)2=02=0.
  • Squaring any number always results in a positive number, so when xx is any value other than 8, (x8)(x8) will be a number other than 0, and when squared, (x8)2(x8)2 will be positive.
  • Any positive number is greater than 0, so when x8x8, the value of (x8)2(x8)2 will be greater than when x=8x=8. In other words, pp has the least value when x=8x=8.

Use similar reasoning to explain why the point (4,1)(4,1) corresponds to the maximum value of qq, defined by q(x)=2(x4)2+1q(x)=2(x4)2+1.

The greatest value of a function is the maximum value. The lowest value of a function is the minimum value. Since infinity cannot be reached or touched, it is not considered a maximum or minimum.

In quadratic functions, the maximum or minimum value occurs at the vertex of the parabola.

2. Here are some quadratic functions and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function.

Equation Coordinates of the vertex
a. f(x)=(x4)2+6f(x)=(x4)2+6 (4,6)(4,6)
b. g(x)=(x+7)23g(x)=(x+7)23 (7,3)(7,3)
c. h(x)=4(x+5)2+7h(x)=4(x+5)2+7 (5,7)(5,7)
d. k(x)=x26x3k(x)=x26x3 (3,12)(3,12)
e. m(x)=x2+8xm(x)=x2+8x (4,16)(4,16)

a. Determine if the vertex corresponds to the maximum or the minimum value of the function f(x)=(x4)2+6f(x)=(x4)2+6.

b. Determine if the vertex corresponds to the maximum or the minimum value of the function g(x)=(x+7)23g(x)=(x+7)23.

c. Determine if the vertex corresponds to the maximum or the minimum value of the function h(x)=4(x+5)2+7h(x)=4(x+5)2+7.

d. Determine if the vertex corresponds to the maximum or the minimum value of the function k(x)=x26x3k(x)=x26x3.

e. Determine if the vertex corresponds to the maximum or the minimum value of the function m(x)=x2+8xm(x)=x2+8x.

Are you ready for more?

Extending Your Thinking

1.

Here is a portion of the graph of function qq, defined by q(x)=x2+14x40q(x)=x2+14x40.

Graph of a parabola on a coordinate plane where the axes are unlabeled. A red rectangle is superimposed over the parabola in the first quadrant.

ABCDABCD is a rectangle. Points AA and BB coincide with the xx-intercepts of the graph, and segment CDCD just touches the vertex of the graph.

Find the area of ABCDABCD. Show your reasoning.

Self Check

Which of the following is the correct vertex for h ( x ) = 3 ( x 2 ) 2 + 5 ? Describe the vertex as a maximum or minimum.
  1. ( 2 , 5 ) , minimum
  2. ( 2 , 5 ) , maximum
  3. ( 2 , 5 ) , minimum
  4. ( 2 , 5 ) , maximum

Additional Resources

Identify a Maximum or Minimum

Recall that the maximum is the highest point of the graph and the minimum is the lowest point of the graph. A quadratic function shape is called a parabola, and a parabola will only have a maximum or a minimum.

Example 1

Graph f(x)=(x2)21f(x)=(x2)21.

Graph of a parabola on a coordinate plane. The vertex (2, negative 1) has been labeled on the parabola. Both the x- and y-axes have a scale of 1 and extend from negative 10 to 10.

In the vertex form (of a quadratic expression), f(x)=a(xh)2+kf(x)=a(xh)2+k, tthe vertex (of a graph) is located at (h,k)(h,k).

For the function graphed, the vertex is at (2,1)(2,1).

Since the leading coefficient, aa, is positive, the parabola opens up. This means the vertex is a minimum.

Example 2

Graph f(x)=(x+1)2+3f(x)=(x+1)2+3.

Graph of a parabola on a coordinate plane. The vertex (negative 1, 3) has been labeled on the parabola. Both the x- and y-axes have a scale of 1 and extend from negative 10 to 10.

The vertex is located at (1,3)(1,3).

Since the leading coefficient is negative, the parabola opens down and the vertex is a maximum.

Try it

Try It: Identify a Maximum or Minimum

For the function f(x)=(x+4)22f(x)=(x+4)22, identify the vertex and tell if it is a maximum or minimum.

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