The quadratic function f is defined by f(x) = (x - 5)(x - 11). In the xy-plane, the graph … — SAT

The quadratic function $f$ is defined by $f(x) = (x - 5)(x - 11)$ . In the $xy$ -plane, the graph of $y = f(x)$ is a parabola with vertex $(h, k)$ . What is the value of $h$ ?

Answer: 8

Answer

The value of

h

8

The given quadratic function is in factored form,

f(x) = (x - 5)(x - 11)

, which shows that the

x

-intercepts of the graph are at

x = 5

and

x = 11

. Because a parabola is symmetric, the

x

-coordinate of its vertex,

h

, is located exactly halfway between the two

x

-intercepts. Calculating the average of the

x

-intercepts gives

h = \frac{5 + 11}{2} = 8

Step-by-Step Solution

Find the

x

-intercepts of the parabola.

The

x

-intercepts are

x = 5

and

x = 11

The

x

-intercepts of a quadratic function in the form

f(x) = (x - r_1)(x - r_2)

occur at

x = r_1

and

x = r_2

because setting

f(x) = 0

yields these roots.

Calculate the

x

-coordinate of the vertex,

h

h = 8

Since a parabola is symmetric, its vertex lies on the axis of symmetry, which is the vertical line halfway between the

x

-intercepts. The average of

5

and

11

\frac{5 + 11}{2} = 8

Key Concept

Finding the vertex of a parabola from its factored form

Question

Answer

Step-by-Step Solution

Key Concept

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