The function f is defined by f(x) = (x - 2)^2 - 5. The graph of the function g in the xy-p… — SAT

The function $f$ is defined by $f(x) = (x - 2)^2 - 5$ . The graph of the function $g$ in the $xy$ -plane is obtained by translating the graph of $f$ by $3$ units to the left and $4$ units up. If the vertex of the graph of $g$ is $(h, k)$ , what is the value of $h + k$ ?

A
-6
-2Answer
C
4
D
-10

Answer

-2

The correct answer is

-2

. The vertex form of a quadratic function is given by

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

, where

(h, k)

is the vertex. For the function

f(x) = (x - 2)^2 - 5

, the vertex is at

(2, -5)

. Translating the graph of

f

3

units to the left shifts the

x

-coordinate of the vertex to

2 - 3 = -1

. Translating the graph of

f

4

units up shifts the

y

-coordinate of the vertex to

-5 + 4 = -1

. Therefore, the vertex of the graph of

g

(h, k) = (-1, -1)

, and the value of

h + k

-1 + (-1) = -2

Step-by-Step Solution

Identify the vertex of the original function

f(x) = (x - 2)^2 - 5

The vertex is

(2, -5)

For a quadratic function in vertex form

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

, the coordinates of the vertex are

(h, k)

Apply the horizontal translation of

3

units to the left to the vertex's

x

-coordinate.

The new

x

-coordinate is

2 - 3 = -1

Shifting a graph horizontally to the left by

c

units subtracts

c

from the

x

-coordinates of its points.

Apply the vertical translation of

4

units up to the vertex's

y

-coordinate.

The new

y

-coordinate is

-5 + 4 = -1

Shifting a graph vertically upward by

c

units adds

c

to the

y

-coordinates of its points.

Calculate the sum of the coordinates of the new vertex

(h, k) = (-1, -1)

h + k = -1 + (-1) = -2

The question asks for the value of the sum

h + k

, where

(h, k)

is the vertex of the graph of

g

Key Concept

Quadratic function vertex transformations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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