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Chapter 5: Problem 35

(a) Use paper and pencil to determine the intercepts and asymptotes for the graph of each function. (b) Use a graphing utility to graph each function. Your results in part (a) will be helpful in choosing an appropriate viewing rectangle that shows the essential features of the graph. $$y=4^{-x}-4$$

Short Answer

Expert verified
The y-intercept is at (0, -3), x-intercept at (-1, 0), and the horizontal asymptote is y = -4.

Step by step solution

01

Finding the y-intercept

To determine the y-intercept of the function \( y=4^{-x}-4 \), set \( x = 0 \). Substituting \( x = 0 \), we have \( y = 4^{0} - 4 = 1 - 4 = -3 \). Thus, the y-intercept is at the point \( (0, -3) \).
02

Finding the x-intercept

To find the x-intercept of the function \( y=4^{-x}-4 \), set \( y = 0 \). Thus, \( 0 = 4^{-x} - 4 \). Solving for \( x \), we get \( 4^{-x} = 4 \), which gives \( -x = 1 \) and consequently, \( x = -1 \). Therefore, the x-intercept is at the point \( (-1, 0) \).
03

Determining the Horizontal Asymptote

The horizontal asymptote of an exponential function \( y = a^{-x} + b \) is determined by the constant term \( b \). Thus, for the function \( y=4^{-x}-4 \), the horizontal asymptote is \( y = -4 \). This is because as \( x \to \infty \), the term \( 4^{-x} \to 0 \), so \( y \to -4 \).
04

Graphing the Function

Using a graphing utility, plot the function \( y = 4^{-x} - 4 \). The key features identified are: the y-intercept at \( (0, -3) \), the x-intercept at \( (-1, 0) \), and the horizontal asymptote at \( y = -4 \). Set the viewing rectangle to include these points and asymptote to appropriately visualize the graph.

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Key Concepts

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

Intercepts
Intercepts are crucial points where a graph intersects an axis. To find the y-intercept of a function, set the variable \( x \) to zero and solve for \( y \). For the equation \( y = 4^{-x} - 4 \), by substituting \( x = 0 \), we obtain:
  • \( y = 4^{0} - 4 = 1 - 4 = -3 \)
This calculation means the graph crosses the y-axis at point \( (0, -3) \), which is the y-intercept.
For the x-intercept, set \( y = 0 \) and solve for \( x \). Thus, for \( 0 = 4^{-x} - 4 \), we get:
  • \( 4^{-x} = 4 \)
  • \( -x = 1 \)
  • \( x = -1 \)
Hence, the graph cuts the x-axis at point \( (-1, 0) \). Understanding these intercepts helps in sketching the basic structure of the graph.
Horizontal Asymptote
A horizontal asymptote is a horizontal line that the graph approaches as \( x \) approaches either positive or negative infinity. It provides insight into the long-term behavior of a function. In exponential functions of the form \( y = a^{-x} + b \), the horizontal asymptote is defined by the constant \( b \).
For our function \( y = 4^{-x} - 4 \), the constant \( b \) is \( -4 \). Therefore, the horizontal asymptote is \( y = -4 \).
This means that as \( x \) becomes very large, the term \( 4^{-x} \) approaches zero, causing the graph to flatten out near the line \( y = -4 \).
Understanding the horizontal asymptote helps us comprehend how the graph behaves at its extremes. It's like a visual boundary that the function will get infinitely close to but never quite reach.
Exponential Equations
Exponential equations are equations in which variables appear as exponents. They often model situations involving growth or decay, such as population dynamics or radioactive decay. The general form is \( y = a^{x} + b \), where:
  • \( a \) is a positive constant representing the base of the exponential.
  • \( x \) is the exponent variable.
  • \( b \) is a constant that translates the graph vertically.
In the function \( y = 4^{-x} - 4 \), the base \( a \) is 4, and the negative exponent \( -x \) suggests a decreasing function as \( x \) increases. This creates a curve moving towards the horizontal asymptote from above.
Exponential equations can be manipulated in solving various problems, making it a versatile mathematical tool.
Graphing Utilities
Graphing utilities are tools or software that help visualize mathematical functions quickly and accurately. They can be physical calculators or computer programs.
For the function \( y = 4^{-x} - 4 \), using a graphing utility makes it easier to plot the intercepts and see the horizontal asymptote. You can also adjust the viewing window to focus on important features like intercepts and asymptotes. Here are steps to efficiently use a graphing utility for this function:
  • Enter the function equation into the graphing tool.
  • Set the viewing rectangle to include points \( (0, -3) \), \( (-1, 0) \), and the line \( y = -4 \).
  • Observe the curve approaching the asymptote at \( y = -4 \) as \( x \) increases.
By providing visual confirmation of calculated results, graphing utilities can enhance understanding and provide insight into complex functions.

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