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Chapter 5: Problem 27

Sketch the graph of the function and check the graph with a graphing calculator. Describe how each graph can be obtained from the graph of a basic exponential function.. $$f(x)=2^{x+1}$$

Short Answer

Expert verified
Shift the graph of \( y = 2^x \) 1 unit to the left.

Step by step solution

01

- Identify the Basic Function

The basic function is an exponential function of the form \( f(x) = a^x \). In this problem, the base of the exponential function is 2. So the basic function is \( f(x) = 2^x \).
02

- Understand the Transformation

The function given is \( f(x) = 2^{x+1} \). This indicates a horizontal shift of the basic function \( f(x) = 2^x \). Specifically, the \(x+1\) inside the exponent will shift the graph to the left by 1 unit.
03

- Sketch the Basic Exponential Function

First, sketch the graph of the basic exponential function \( y = 2^x \). This graph passes through the points (0, 1), (1, 2), and (-1, 0.5), and it increases rapidly.
04

- Apply the Transformation

Since the function is \( f(x) = 2^{x+1} \), shift the graph of \( y = 2^x \) 1 unit to the left. This means each point \((x, y)\) on the graph of \( y = 2^x \) is transformed to \((x-1, y)\).
05

- Check With a Graphing Calculator

Use a graphing calculator to input \( f(x) = 2^{x+1} \). Compare the calculator's graph with your hand-drawn graph to ensure accuracy. Both should show the graph shifted left by 1 unit.
06

- Describe the Transformation

The graph of \( f(x) = 2^{x+1} \) can be obtained by shifting the graph of the basic exponential function \( y = 2^x \) to the left by 1 unit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Basic Exponential Function
An exponential function is a mathematical expression where a constant base is raised to a variable exponent, generally represented as \( f(x) = a^x \). In this basic form, the function grows (or decays) rapidly:
  • For bases greater than 1 (e.g., \( a = 2 \)), the function grows exponentially as \( x \) increases.
  • For bases between 0 and 1, the function decays exponentially.
One fundamental exponential function is \( f(x) = 2^x \). This function:
- Passes through the point (0, 1), since any number to the power of 0 is 1.
  • Includes points like (1, 2) and (-1, 0.5), representing rapid growth and decay respectively.
  • The graph has a horizontal asymptote at \( y = 0 \); it approaches zero but never touches it.
Graphing this basic function is the first step in understanding more complex transformations.
Horizontal Shift
A horizontal shift moves the graph left or right along the x-axis. For the function \( f(x) = 2^{x+1} \), the expression \( x+1 \) signifies that we shift the graph to the left by 1 unit. Why the move to the left? In general:
  • When the exponent has a positive constant added (\( x + k \)), we shift left by \( k \) units.
  • If the exponent has a negative constant added (\( x - k \)), we shift right by \( k \) units.
Applying this to the given function:
- Take the basic exponential function \( y = 2^x \) which passes through (0, 1), (1, 2), and (-1, 0.5).
  • To shift left 1 unit, subtract 1 from each x-coordinate.
  • New points become (-1, 1), (0, 2), and (-2, 0.5), reflecting the leftward shift.
Visualize this step to confirm the understanding of the horizontal transformation.
Graph Transformation
Graph transformation applies when you need to modify the base graph to match a transformed function. Steps to transform \( y = 2^x \) into \( f(x) = 2^{x+1} \):
1. Start with the graph of \( y = 2^x \).
2. Apply the horizontal shift: Shift each point left by 1 unit (i.e., change (x, y) to (x-1, y)).
Check these points:
  • (0, 1) moves to (-1, 1)
  • (1, 2) shifts to (0, 2)
  • (-1, 0.5) transforms to (-2, 0.5)
Using a graphing calculator can help verify the transformation. Input \( f(x) = 2^{x+1} \) and compare with your hand-drawn graph to ensure they match.

Buy Useful Pointers: Practice is key! Try shifting graphs of different exponential functions and observe the transformations. Use graphing tools for a quick visual comparison to reinforce your understanding.

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