A transformation is applied to the exponential function f(x) = 2^x - 3 to obtain a new fun… — SAT

A transformation is applied to the exponential function $f(x) = 2^x - 3$ to obtain a new function $g$ , where $g(x) = -f(x - 1) + 2$ . What is the $y$ -intercept of the graph of $y = g(x)$ in the $xy$ -plane?

A
$(0, 3)$
B
$(0, -\frac{3}{2})$
$(0, \frac{9}{2})$ Answer
D
$(0, -\frac{1}{2})$

Answer

(0, \frac{9}{2})

The correct answer is the coordinate point with

y

-coordinate

\frac{9}{2}

. The

y

-intercept of the graph of

g(x)

is found by evaluating

g(0)

. Substituting

x = 0

into the transformation equation gives

g(0) = -f(-1) + 2

. Using the definition of

f(x)

, we find

f(-1) = 2^{-1} - 3 = 0.5 - 3 = -2.5

. Substituting this value back into the equation for

g(0)

yields

g(0) = -(-2.5) + 2 = 2.5 + 2 = 4.5

, which is equivalent to

\frac{9}{2}

. Therefore, the

y

-intercept is the coordinate point containing

\frac{9}{2}

Step-by-Step Solution

Substitute

x = 0

into the definition of

g(x)

to find the

y

-coordinate of the

y

-intercept.

g(0) = -f(0 - 1) + 2 = -f(-1) + 2

The

y

-intercept of any function

y = g(x)

occurs where the input

x

is equal to

0

Evaluate

f(-1)

using the definition

f(x) = 2^x - 3

f(-1) = 2^{-1} - 3 = \frac{1}{2} - 3 = -2.5

Evaluating

f(-1)

is necessary to substitute its value back into the expression for

g(0)

Substitute the value of

f(-1)

back into the expression for

g(0)

and simplify.

g(0) = -(-2.5) + 2 = 2.5 + 2 = 4.5 = \frac{9}{2}

This yields the final

y

-coordinate of the

y

-intercept, which is

\frac{9}{2}

, giving the coordinate point

(0, \frac{9}{2})

Key Concept

Applying horizontal translation, reflection across the x-axis, and vertical translation to evaluate a function at a specific point.

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