A quadratic function g is defined by g(x) = 3(x - 5)^2 - 4. If the graph of y = g(x) in th… — SAT

A quadratic function $g$ is defined by $g(x) = 3(x - 5)^2 - 4$ . If the graph of $y = g(x)$ in the $xy$ -plane is translated $6$ units up to produce the graph of $y = f(x)$ , what is the $y$ -coordinate of the vertex of the graph of $y = f(x)$ ?

Answer: 2

Answer

The correct answer is 2.

The original function

g(x) = 3(x - 5)^2 - 4

is in vertex form

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

, where the vertex is

(h, k)

. Therefore, the vertex of the graph of

g

(5, -4)

. Translating a graph upward by

6

units is represented by adding

6

to the function, so

f(x) = g(x) + 6

. This transformation shifts the vertex from

(5, -4)

(5, -4 + 6)

, which is

(5, 2)

. The

y

-coordinate of this new vertex is

2

Step-by-Step Solution

Identify the vertex of the original quadratic function.

The vertex of the graph of

g(x) = 3(x - 5)^2 - 4

(5, -4)

A quadratic function written in vertex form

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

has its vertex at

(h, k)

Determine the vertex of the translated function.

The vertex of the graph of

y = f(x)

(5, 2)

Translating a graph

6

units up increases the

y

-coordinate of all points, including the vertex, by

6

Key Concept

Vertex form of a quadratic function and vertical translation of graphs.

Estimated Time:45s

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