In the xy-plane, the graph of y = x^2 is shifted 4 units to the right and 9 units up to pr… — SAT

In the $xy$ -plane, the graph of $y = x^2$ is shifted $4$ units to the right and $9$ units up to produce the graph of $y = h(x)$ . Which of the following equations defines the function $h$ ?

A
$h(x) = (x + 4)^2 + 9$
B
$h(x) = (x - 4)^2 - 9$
$h(x) = (x - 4)^2 + 9$ Answer
D
$h(x) = (x + 4)^2 - 9$

Answer

h(x) = (x - 4)^2 + 9

The parent function

y = x^2

has its vertex at

(0, 0)

. Shifting this graph

4

units to the right moves the x-coordinate of the vertex to

4

, and shifting it

9

units up moves the y-coordinate of the vertex to

9

. The new vertex is at

(4, 9)

. A parabola with a vertex of

(h, k)

and a leading coefficient of

1

is written in vertex form as

y = (x - h)^2 + k

. Substituting

h = 4

and

k = 9

gives the equation

h(x) = (x - 4)^2 + 9

Step-by-Step Solution

Identify the base function and the rules of graph translation.

The base function is

y = x^2

. Shifting a graph horizontally by

h

units changes the input

x

(x - h)

, and shifting vertically by

k

units adds

k

to the function value.

Applying transformations to the parent function changes its position in the coordinate plane while maintaining its shape.

Apply the horizontal translation of

4

units to the right.

Replacing

x

with

(x - 4)

in the parent function gives the intermediate equation

y = (x - 4)^2

A horizontal shift of

h

units to the right corresponds to replacing the input variable

x

with

(x - h)

Apply the vertical translation of

9

units up.

Adding

9

to the expression yields the final equation

h(x) = (x - 4)^2 + 9

A vertical shift of

k

units upward corresponds to adding

k

to the output of the function.

Key Concept

Quadratic graph transformations and translation rules

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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