In the xy-plane, the graph of the quadratic function f has its vertex at the point (4, -3)… — SAT

In the $xy$ -plane, the graph of the quadratic function $f$ has its vertex at the point $(4, -3)$ . The function $g$ is defined by $g(x) = f(x + 2) + 5$ . Which of the following ordered pairs represents the vertex of the graph of $g$ ?

A
$(6, 2)$
B
$(2, -8)$
$(2, 2)$ Answer
D
$(6, -8)$

Answer

(2, 2)

The vertex of

f(x)

is given as

(4, -3)

. The translation

g(x) = f(x + 2) + 5

shifts the graph

2

units to the left and

5

units upward. Applying these transformations to the vertex of

f(x)

yields the new vertex

(4 - 2, -3 + 5) = (2, 2)

Step-by-Step Solution

Identify the vertex of the function

f(x)

The vertex of

f(x)

(4, -3)

This is given in the problem and serves as the starting point for the transformations.

Determine the effect of the horizontal translation

f(x + 2)

The vertex shifts

2

units to the left, changing the

x

-coordinate from

4

4 - 2 = 2

For any function

f(x)

, the graph of

f(x + c)

is shifted

c

units to the left when

c > 0

Determine the effect of the vertical translation

+ 5

The vertex shifts

5

units upward, changing the

y

-coordinate from

-3

-3 + 5 = 2

Adding a constant to a function shifts its graph vertically upward by that constant's value.

Combine the translated coordinates to find the vertex of

g(x)

The vertex of the graph of

g(x)

(2, 2)

The horizontal shift results in an

x

-coordinate of

2

, and the vertical shift results in a

y

-coordinate of

2

Key Concept

Quadratic Transformations and Vertex Shifts

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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