Answered: The graph f(x)=3^x is reflected across…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter2: Functions

Section: Chapter Questions

Problem 30P: In this problem you are asked to find a function that models in real life situation and then use the...

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Question

The graph f(x)=3^x is reflected across the cacos shifted downward 5 units and the. Shifted left 2 units. Let g(x) represent the new function. What is an equation for this function? G(x) =? What is the y-intercept of g(x) enter as ordered pair. What is the domain of g(x)? Enter as interval. What is the range of g(x) enter as interval.

Expert Solution

Step 1

Given:

The graph of $f (x) = 3^{x}$ is reflected across the x axis, shifted downward 5 units and then shifted left 2 units.

Reflection across x axis:

To reflect the graph of the function $y = f (x)$ about the x axis, multiply $f (x)$ by $- 1$ .

Thus, the graph of the function $y = - f (x)$ is the reflection of the graph of $y = f (x)$ about the x axis.

Vertical shift:

The graph of $y = f (x) - h$ is the graph of $y = f (x)$ shifted down h units where $h > 0$ .

Horizontal shift:

The graph of $y = f (x + h)$ is the horizontal shift of graph of $y = f (x)$ to the left by h units where $h > 0$ .

Step 2

By the above definitions, the graph of $y = - 3^{x}$ is the reflection of the graph of $y = 3^{x}$ across the x axis.

The graph of $y = - 3^{x} - 5$ is the graph of $y = - 3^{x}$ shifted down 5 units.

The graph of $y = - 3^{x + 2} - 5$ is the graph of $y = - 3^{x} - 5$ shifted to the left by 2 units.

Therefore, the new function is $y = - 3^{x + 2} - 5$ .

Let the new function be $g (x)$ .

Therefore, the equation for this function is $g (x) = - 3^{x + 2} - 5$ .

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