/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 835 Construct the graph of \(\mathrm... [FREE SOLUTION] | 91影视

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Chapter 26: Problem 835

Construct the graph of \(\mathrm{y}=3^{\mathrm{x}}\).

Short Answer

Expert verified
To construct the graph of \(y = 3^x\), plot the following points on a coordinate plane: \((-2, 0.111), (-1, 0.333), (0, 1), (1, 3), (2, 9)\). Connect the points keeping the characteristics of an exponential function in mind: it passes through the point (0,1), is always above the x-axis, and increases rapidly as x increases while approaching the x-axis as x decreases. The resulting graph will show a typical exponential growth pattern.

Step by step solution

01

Understanding the exponential function

An exponential function is a function of the form \( y = a^x \) where \( a > 0 \) and \( a \neq 1 \). In our case, \( a = 3 \) and the function is \( y = 3^x \). The graph of an exponential function has the following characteristics: 1. For \( x = 0 \), \( y = a^0 = 1 \), so the graph passes through the point (0, 1). 2. The graph is always above the x-axis, i.e., \( y > 0 \). 3. As x approaches infinity, y approaches infinity. And as x approaches negative infinity, y approaches 0.
02

Choose some points to plot

Based on the characteristics of an exponential function, we can choose the following x-values to plot: \( -2, -1, 0, 1, 2 \). For each of these x-values, we compute the corresponding y-values: 1. For \( x = -2 \), \( y = 3^{-2} = \frac{1}{9} \approx 0.111 \). 2. For \( x = -1 \), \( y = 3^{-1} = \frac{1}{3} \approx 0.333 \). 3. For \( x = 0 \), \( y = 3^0 = 1 \). 4. For \( x = 1 \), \( y = 3^1 = 3 \). 5. For \( x = 2 \), \( y = 3^2 = 9 \). These give us the points: \( (-2, 0.111), (-1, 0.333), (0, 1), (1, 3), (2, 9) \).
03

Plotting the points and graphing the function

Plot the chosen points on a coordinate plane: 1. (-2, 0.111) 2. (-1, 0.333) 3. (0, 1) 4. (1, 3) 5. (2, 9) Now, connect the points to create the graph of the function \( y = 3^x \), keeping in mind the characteristics of the exponential function. The graph will increase rapidly as x increases, and will approach the x-axis but never touch it as x decreases. The resulting graph of the function \( y = 3^x \) will show a typical exponential growth pattern.

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