The graph of the function f in the xy-plane has a vertex at (-2, 7). The function g is def… — SAT

The graph of the function $f$ in the $xy$ -plane has a vertex at $(-2, 7)$ . The function $g$ is defined by $g(x) = f(-x + 3) + 12$ . If the vertex of the graph of $y = g(x)$ is the point $(a, b)$ , what is the value of $a + b$ ?

Answer: 24

Answer

The vertex of the parent function

f(x)

is at

(-2, 7)

, which means

f(-2) = 7

. The transformed function is

g(x) = f(-x + 3) + 12

. The vertex of

g(x)

occurs when the input to

f

, which is

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

gives

-x = -5

, or

x = 5

, so the x-coordinate of the vertex of

g(x)

a = 5

. To find the y-coordinate

b

, we evaluate

g(5) = f(-(5) + 3) + 12 = f(-2) + 12 = 7 + 12 = 19

. Therefore, the vertex of the graph of

y = g(x)

(5, 19)

, so

a = 5

and

b = 19

. The value of

a + b

5 + 19 = 24

Step-by-Step Solution

Find the x-coordinate of the vertex of the transformed function

g(x)

by setting the inner expression equal to the x-coordinate of the original vertex.

a = 5

The vertex of

f(x)

is located at

x = -2

. For

g(x) = f(-x + 3) + 12

, the vertex occurs when the input to

f

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

yields

x = 5

Find the y-coordinate of the vertex of

g(x)

by evaluating

g(5)

b = 19

Substituting

x = 5

into the definition of

g(x)

gives

g(5) = f(-2) + 12

. Since the vertex of

f

is at

(-2, 7)

f(-2) = 7

. Thus,

g(5) = 7 + 12 = 19

Calculate the sum of the coordinates

a

and

b

The vertex of

g(x)

(5, 19)

, so

a = 5

and

b = 19

. The sum

a + b

5 + 19 = 24

Key Concept

Determining the coordinates of a transformed vertex using function notation.

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