For the function f(x) = 4^x - 3, a new function g is created by reflecting the graph of y … — SAT

For the function $f(x) = 4^x - 3$ , a new function $g$ is created by reflecting the graph of $y = f(x)$ across the $x$ -axis, followed by a vertical translation of $c$ units upward. The graph of $y = g(x)$ contains the point $(2, -5)$ . What is the value of the constant $c$ ?

Answer: 8

Answer

Reflecting the graph of

f(x) = 4^x - 3

across the

x

-axis results in the function

-f(x) = -(4^x - 3) = -4^x + 3

. Translating this graph vertically upward by

c

units defines

g(x) = -4^x + 3 + c

. Substituting the given point

(2, -5)

yields

-5 = -4^2 + 3 + c

, which simplifies to

-5 = -13 + c

. Adding

13

to both sides gives

c = 8

Step-by-Step Solution

Reflect the function

f(x)

across the

x

-axis

-f(x) = -(4^x - 3) = -4^x + 3

Reflecting a graph across the

x

-axis is represented by negating the output of the function, which transforms

y = f(x)

y = -f(x)

Apply the vertical translation upward by

c

units to define

g(x)

g(x) = -f(x) + c = -4^x + 3 + c

Translating a function vertically upward by

c

units adds

c

to the function's expression.

Substitute the point

(2, -5)

into

g(x)

and solve for

c

c = 8

Since the graph of

g

passes through

(2, -5)

, substitute

x = 2

and

g(x) = -5

into the equation:

-5 = -4^2 + 3 + c

. This simplifies to

-5 = -16 + 3 + c

, then

-5 = -13 + c

, which gives

c = 8

Key Concept

Function transformations, including vertical reflections and translations

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