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Chapter 5: Problem 18

Graph the pair of functions on the same set of axes. $$y=3^{x} ; y=3^{-x}$$

Short Answer

Expert verified
Graph the functions, intersecting the y-axis at (0, 1), with \( y = 3^x \) showing growth and \( y = 3^{-x} \) showing decay.

Step by step solution

01

Understand the Function Types

The functions given are exponential. The first function, \( y = 3^x \), is an exponential growth function, while the second function, \( y = 3^{-x} \), is an exponential decay function. Both use base 3.
02

Create a Table for \( y = 3^x \)

Choose a set of x-values (e.g., -2, -1, 0, 1, 2) and calculate their corresponding y-values for the first function. For example:\[\begin{array}{c|c}x & y = 3^x \ \hline-2 & \frac{1}{9} \-1 & \frac{1}{3} \0 & 1 \1 & 3 \2 & 9 \\end{array}\]
03

Create a Table for \( y = 3^{-x} \)

Repeat the process for the second function with the same x-values:\[\begin{array}{c|c}x & y = 3^{-x} \ \hline-2 & 9 \-1 & 3 \0 & 1 \1 & \frac{1}{3} \2 & \frac{1}{9} \\end{array}\]
04

Plot the Points for \( y = 3^x \)

Using the table from Step 2, plot the points on the graph. These points are (-2, 1/9), (-1, 1/3), (0, 1), (1, 3), and (2, 9). Connect these points smoothly to show exponential growth.
05

Plot the Points for \( y = 3^{-x} \)

Now plot the points from the table in Step 3: (-2, 9), (-1, 3), (0, 1), (1, 1/3), and (2, 1/9). Connect these points smoothly to depict exponential decay.
06

Finalize the Graph

Ensure both functions are on the same set of axes and labeled clearly. Note that \( y = 3^x \) starts low and increases rapidly, while \( y = 3^{-x} \) starts high and decreases. Both intersect the y-axis at (0, 1).

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Key Concepts

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

Exponential Growth
In mathematics, exponential growth refers to a situation where the quantity increases rapidly over time. The function \( y = 3^x \) is a classic example of exponential growth. Here, the base of the exponential function is 3, which means each increase of 1 in the value of \( x \) results in the value of \( y \) being multiplied by 3.

Exponential growth functions are characterized by:
  • A rapid increase in the value of \( y \) as \( x \) becomes larger.
  • Y-values that are positive for all real numbers \( x \).
  • A curve that moves upwards steeply as it progresses from left to right.
  • A horizontal asymptote at y = 0, meaning as \( x \) approaches negative infinity, \( y \) approaches zero.
Understanding exponential growth is crucial in various fields, such as biology, finance, and physics, where quantities increase profoundly in a short period, like population growth or compounding interest.
Exponential Decay
Exponential decay describes a process where the value decreases rapidly as time goes on. The function \( y = 3^{-x} \) represents exponential decay. Here, the negative exponent signifies that as \( x \) increases, the value of \( y \) diminishes at an accelerating rate.

Some key characteristics of exponential decay include:
  • The value of \( y \) shrinks toward zero as \( x \) gets larger.
  • The graph is a mirrored version of exponential growth across the y-axis.
  • It features a downward slope with the curve flattening as it approaches zero.
  • Similarly, it also has a horizontal asymptote at y = 0.
Exponential decay is often seen in natural phenomena like radioactive decay, cooling of objects, or depreciation of assets. Understanding this concept helps in predicting how quickly a quantity will reduce over time.
Graphing Functions
Graphing functions is a powerful way to visualize and understand mathematical relationships. For the functions \( y = 3^x \) and \( y = 3^{-x} \), the graph shows us clear distinctions between exponential growth and decay.

Graphing involves these steps:
  • Choose a reasonable range of \( x \)-values. In our case, using -2, -1, 0, 1, 2 helps visualize both growth and decay.
  • Calculate corresponding \( y \)-values, which are gained from substituting each \( x \)-value into the function equation.
  • Plot each point (\( x, y \)) on a Cartesian plane for both functions.
  • Connect the points smoothly to reveal the function's curve.
Notice that both functions intersect the y-axis at (0, 1). This intersection represents the fact that any non-zero number raised to the power of 0 equals 1. Graphing helps identify such insightful points and provides a visual interpretation of how functions behave over different intervals.

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Most popular questions from this chapter

Solve each equation. $$3(\ln x)^{2}-\ln \left(x^{2}\right)-8=0$$

Solve the inequalities. Where appropriate, give an exact answer as well as a decimal approximation. $$e^{x^{2}-4 x} \geq e^{5}$$

Solve each equation. $$x^{1+\log _{x} 16}=4 x^{2}$$

Solve the inequality \(\log _{2} x+\log _{2}(x+1)-\log _{2}(2 x+6)<0.\)

(a) Let \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t} .\) Show that \([\mathcal{N}(t+1)-\mathcal{N}(t)] / \mathcal{N}(t)=e^{k}-1 .\) (This is actually done in detail in the text. So, ideally, you should look back only if you get stuck or want to check your answer.) (b) Assume as given the following approximation, which was introduced in Exercise 26 of Section 5.2 \(e^{x} \approx x+1 \quad\) provided \(x\) is close to zero Use this approximation to explain why \(e^{k}-1 \approx k\) provided that \(k\) is close to zero. Remark: Combining this result with that in part (a), we conclude that the relative growth rate for the function \(\mathcal{N}(t)=\mathcal{N}_{0} e^{k t}\) is approximately equal to the growth constant \(k .\) As explained in the text, this is one of the reasons why in applications we've not distinguished between the relative growth rate and the decay constant \(k\)
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