Sketch the graph of the function and check the graph with a graphing calculator. Before doing so, describe how the graph of the function can be obtained from the graph of a basic exponential function.
[tex]f(x)=e^{-x-4}[/tex]
Shift the graph of [tex]y =((e))^{ x }[/tex] left 1 unit(s) and then reflect it across the [tex]y[/tex]-axis.
(Type an exact answer in terms of [tex]e[/tex].)
Use the graphing tool to graph the equation.
Answer (2)
Rewrite the function as f ( x ) = e − ( x + 4 ) .
Shift the graph of y = e x to the left by 4 units to obtain y = e x + 4 .
Reflect the graph of y = e x + 4 across the y-axis to obtain y = e − ( x + 4 ) = e − x − 4 .
The graph of f ( x ) = e − x − 4 is obtained by shifting the graph of y = e x left by 4 units and then reflecting it across the y-axis. The shift is 4 units, not 1. The final answer is 4. $
\boxed{4} $
Explanation
Problem Analysis We are given the function f ( x ) = e − x − 4 and asked to describe how its graph can be obtained from the graph of the basic exponential function y = e x . We also need to sketch the graph.
Rewriting the Function First, let's rewrite the function to make the transformations clearer: f ( x ) = e − x − 4 = e − ( x + 4 ) . This form suggests a horizontal shift and a reflection.
Horizontal Shift To obtain the graph of f ( x ) = e − ( x + 4 ) from y = e x , we first shift the graph of y = e x to the left by 4 units. This gives us the graph of y = e x + 4 .
Reflection across y-axis Next, we reflect the graph of y = e x + 4 across the y-axis. This replaces x with − x , resulting in the graph of y = e − ( x + 4 ) = e − x − 4 , which is the desired function f ( x ) .
Summary of Transformations Therefore, the graph of f ( x ) = e − x − 4 is obtained by shifting the graph of y = e x left by 4 units and then reflecting it across the y-axis.
Correct Transformations The prompt incorrectly suggests shifting the graph of y = e x left by 1 unit and then reflecting it across the y-axis. The correct transformation is a shift left by 4 units followed by a reflection across the y-axis.
Final Answer The final answer is to shift the graph of y = e x left by 4 units and then reflect it across the y-axis.
Examples
Understanding transformations of functions is crucial in many fields. For example, in signal processing, shifting a signal in time corresponds to a horizontal shift of its graph. Reflecting a signal corresponds to flipping its graph. These transformations are used to analyze and manipulate signals in various applications, such as audio processing and image recognition. By understanding how basic functions transform, engineers can design filters and algorithms to achieve specific signal processing goals.
The graph of the function f ( x ) = e − x − 4 is obtained by shifting the graph of y = e x to the left by 4 units and then reflecting it across the y-axis. The transformations produce the correct graph for the given function. Using a graphing calculator will help confirm these transformations are represented accurately in the graph.
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