In the xy-plane, the graph of the quadratic function f has a vertex at (h, k), where h and… — SAT

In the $xy$ -plane, the graph of the quadratic function $f$ has a vertex at $(h, k)$ , where $h$ and $k$ are constants. The graph passes through the points $(1, 10)$ and $(7, 10)$ . If the minimum value of the function $f$ is $1$ , what is the value of $f(2)$ ?

Answer: 5

Answer

The value of

f(2)

is 5.

The correct answer is 5. Since the graph of the quadratic function passes through

(1, 10)

and

(7, 10)

, the axis of symmetry is the vertical line halfway between

x = 1

and

x = 7

, which is

x = \frac{1+7}{2} = 4

. The minimum value of the function is

1

, which occurs at the vertex, so the vertex is

(4, 1)

. In vertex form, the function is

f(x) = a(x-4)^2 + 1

. Substituting the point

(1, 10)

yields

10 = a(1-4)^2 + 1

, which simplifies to

9 = 9a

, so

a = 1

. The function is

f(x) = (x-4)^2 + 1

. Evaluating this function at

x = 2

gives

f(2) = (2-4)^2 + 1 = 5

Step-by-Step Solution

Find the

x

-coordinate of the vertex using the symmetry of the parabola.

h = 4

A parabola is symmetric about its vertical axis of symmetry. Since the points

(1, 10)

and

(7, 10)

have the same

y

-coordinate, the axis of symmetry is exactly halfway between their

x

-coordinates:

x = \frac{1 + 7}{2} = 4

Find the vertex coordinates using the minimum value of the function.

Vertex is

(4, 1)

The vertex of a quadratic function with a minimum value lies on the axis of symmetry, and its

y

-coordinate is the minimum value. Thus, the vertex

(h, k)

(4, 1)

Write the vertex form of the quadratic function and solve for the leading coefficient

a

f(x) = (x - 4)^2 + 1

Using the vertex form

f(x) = a(x - h)^2 + k

, we substitute

h = 4

and

k = 1

to get

f(x) = a(x - 4)^2 + 1

. Substituting the point

(1, 10)

gives

10 = a(1 - 4)^2 + 1 \implies 9 = 9a \implies a = 1

Evaluate

f(2)

using the determined function.

f(2) = 5

Substitute

x = 2

into the equation

f(x) = (x - 4)^2 + 1

to obtain

f(2) = (2 - 4)^2 + 1 = 4 + 1 = 5

Key Concept

Using symmetry and the vertex form of a quadratic function to determine its equation and evaluate values.

Estimated Time:1m 30s

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