Question

Graph the given functions on a common screen. How are these graphs related?

7. \(y = {{\bf{3}}^x}\), \(y = {\bf{1}}{{\bf{0}}^x}\), \(y = {\left( {\frac{{\bf{1}}}{{\bf{3}}}} \right)^x}\), \(y = {\left( {\frac{{\bf{1}}}{{{\bf{10}}}}} \right)^x}\)

Short answer

The graph of the given functions on a common screen is shown below:

The graph of \(y = {3^x}\) is the reflection of the graph \(y = {\left( {\frac{1}{3}} \right)^x}\), and the graph of \(y = {10^x}\) is the reflection of the graph \(y = {\left( {\frac{1}{{10}}} \right)^x}\). The graph of \(y = {10^x}\) increases more rapidly than \(y = {3^x}\), and as \(x\) tends to \( - \infty \) the curves approach 0 faster.

Step-by-step solution

Graph the given functions

The procedure to draw the graph of the given functions on a common screen by using the graphing calculator is as follows:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\({3^X}\)in the\({Y_1}\)tab.
2. Then enter the equation\({\left( {\frac{1}{3}} \right)^X}\)in the\({Y_2}\)tab.
3. Then enter the equation\({10^X}\)in the\({Y_3}\)tab.
4. Then enter the equation\({\left( {\frac{1}{{10}}} \right)^X}\)in the\({Y_4}\)tab.
5. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graphs of the functions\(y = {3^x}\),\(y = {\left( {\frac{1}{3}} \right)^x}\), \(y = {10^x}\), and \(y = {\left( {\frac{1}{{10}}} \right)^x}\) is shown below:

Obtain the relation between the graphs

From the above graph, it is observed that about the \(y\)-axis, the graph of \(y = {3^x}\) is the reflection of the graph \(y = {\left( {\frac{1}{3}} \right)^x}\). Also, about the \(y\)-axis, the graph of \(y = {10^x}\) is the reflection of the graph \(y = {\left( {\frac{1}{{10}}} \right)^x}\).

For the function \(y = {3^x}\), and \(y = {10^x}\), the base of the functions\(y = {3^x}\), and\(y = {10^x}\)is greater than 1, so the graphincreases. The base of the functions\(y = {\left( {\frac{1}{3}} \right)^x}\), and\(y = {\left( {\frac{1}{{10}}} \right)^x}\)is less than 1, so the graphdecreases.

Also, as \(x\) tends to \( - \infty \) the curves approach 0 faster.