The graph of the quadratic function h has a vertex at (2, -5) in the xy-plane. If the func… — SAT

The graph of the quadratic function $h$ has a vertex at $(2, -5)$ in the $xy$ -plane. If the function $g$ is defined by $g(x) = h(x + 3) + 4$ , what is the vertex of the graph of $g$ ?

$(-1, -1)$ Answer
B
$(5, -1)$
C
$(-5, -1)$
D
$(1, -1)$

Answer

The vertex of the graph of

g

(-1, -1)

The vertex of the graph of

h

is given as

(2, -5)

. The function

g(x) = h(x + 3) + 4

represents a transformation of the graph of

h

. Specifically,

h(x + 3)

shifts the graph horizontally to the left by

3

units, and adding

4

shifts the graph vertically up by

4

units. Applying these shifts to the original vertex coordinates: the new

x

-coordinate is

2 - 3 = -1

, and the new

y

-coordinate is

-5 + 4 = -1

. Therefore, the vertex of the graph of

g

(-1, -1)

Step-by-Step Solution

Identify the vertex of the original function

h

The vertex of

h

(2, -5)

The problem states that the graph of the quadratic function

h

has a vertex at

(2, -5)

Determine the horizontal translation from the expression

h(x + 3)

A horizontal shift of

3

units to the left.

Replacing

x

with

x + c

in a function

f(x)

shifts the graph left by

c

units. Here,

c = 3

, so we shift the vertex

x

-coordinate left by

3

2 - 3 = -1

Determine the vertical translation from the addition of

4

A vertical shift of

4

units up.

Adding a constant

d

to a function shifts its graph vertically by

d

units. Here,

d = 4

, so we shift the vertex

y

-coordinate up by

4

-5 + 4 = -1

Combine the shifted coordinates to find the new vertex.

The new vertex is

(-1, -1)

Combining the translated

x

-coordinate of

-1

and

y

-coordinate of

-1

gives the vertex of the graph of

g

Key Concept

Quadratic Function Transformations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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