Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Feb 05, 2024

Identify the horizontal and vertical shifts for the exponential function $f(x) = \frac{1}{2} \cdot (3)^{x+1} + 4$ .

STEP 1

Assumptions
1. The given function is an exponential function of the form $f(x) = a \cdot b^{x-h} + k$ , where $a$ is the vertical stretch/compression factor, $b$ is the base of the exponential, $h$ is the horizontal shift, and $k$ is the vertical shift.
2. The function provided is $f(x) = \frac{1}{2} \cdot (3)^{x+1} + 4$ .

STEP 2

Identify the base form of the exponential function for comparison.
The base form of an exponential function with horizontal and vertical shifts is: $f(x) = a \cdot b^{x-h} + k$

STEP 3

Compare the given function to the base form to identify the horizontal shift $h$ .
The given function is: $f(x) = \frac{1}{2} \cdot (3)^{x+1} + 4$ This can be compared to the base form: $f(x) = a \cdot b^{x-h} + k$

STEP 4

Identify the horizontal shift $h$ from the exponent in the given function.
The exponent in the given function is $x+1$ , which can be rewritten as $x-(-1)$ . This indicates a horizontal shift of $h = -1$ , which means the graph is shifted to the left by 1 unit.

STEP 5

Identify the vertical shift $k$ from the constant term outside the exponential in the given function.
The constant term in the given function is $+4$ , which indicates a vertical shift of $k = 4$ , meaning the graph is shifted up by 4 units.

STEP 6

Combine the identified horizontal and vertical shifts to determine the correct answer.
The horizontal shift is left 1 unit, and the vertical shift is up 4 units.
The correct answer is: The horizontal shift is left 1 and the vertical shift is up 4.

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