The function f is defined by f(x) = -(\sqrt{x+2} - 1). The graph of the function g in the … — SAT

The function $f$ is defined by $f(x) = -(\sqrt{x+2} - 1)$ . The graph of the function $g$ in the $xy$ -plane is obtained by shifting the graph of $f$ to the right by $3$ units and then down by $2$ units. If the point $(a, -5)$ lies on the graph of $g$ , what is the value of $a$ ?

A
5
B
11
17Answer
D
37

Answer

The correct answer is

17

. Shifting the graph of

f(x) = -(\sqrt{x+2} - 1)

to the right by

3

units is represented by

f(x-3) = -(\sqrt{x-1} - 1) = -\sqrt{x-1} + 1

. Shifting this result down by

2

units gives the function

g(x) = f(x-3) - 2 = -\sqrt{x-1} - 1

. Substituting the point

(a, -5)

into the equation for

g(x)

yields

-\sqrt{a-1} - 1 = -5

. Adding

1

to both sides gives

-\sqrt{a-1} = -4

, or

\sqrt{a-1} = 4

. Squaring both sides results in

a - 1 = 16

. Finally, adding

1

to both sides gives

a = 17

Step-by-Step Solution

Apply the horizontal translation to the function

f(x)

f(x-3) = -(\sqrt{x-3+2} - 1) = -(\sqrt{x-1} - 1) = -\sqrt{x-1} + 1

Shifting a function

f(x)

to the right by

3

units replaces

x

with

x-3

Apply the vertical translation to find the expression for

g(x)

g(x) = f(x-3) - 2 = -\sqrt{x-1} + 1 - 2 = -\sqrt{x-1} - 1

Shifting a graph down by

2

units subtracts

2

from the entire function expression.

Substitute the point

(a, -5)

into the equation for

g(x)

and solve for

a

-\sqrt{a-1} - 1 = -5 \implies -\sqrt{a-1} = -4 \implies \sqrt{a-1} = 4 \implies a-1 = 16 \implies a = 17

Since the point

(a, -5)

lies on the graph of

g

, we set

g(a) = -5

and solve for

a

by isolating the radical and squaring both sides.

Key Concept

Function transformations involve substituting

x-h

for

x

for horizontal translations and adding

k

to the function for vertical translations.

Estimated Time:1m 30s

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