Question

chapter 1: exponential functions > section exercises 1.6 > exercise 9
tell whether the function $y = 6^x$ represents exponential growth or exponential decay.
this function represents exponential dropdown
identify the graph of the function.
four graphs are shown, two growth - like and two decay - like, with grids and axes labeled with x from -4 to 4 and y from 0 to 8 approximately

Explanation:

Step1: Recall Exponential Function Form

The general form of an exponential function is \( y = a b^x \), where \( a>0 \) and \( b>0 \), \( b
eq1 \). If \( b > 1 \), it's exponential growth; if \( 0 < b < 1 \), it's exponential decay.
For \( y = 6^x \), here \( a = 1 \) and \( b = 6 \). Since \( 6>1 \), this is exponential growth.

Step2: Analyze Graphs for Exponential Growth

Exponential growth graphs increase as \( x \) increases (rise from left to right), and pass through \( (0, a) \) (here \( (0, 1) \), but scaled by \( a \); for \( y = 6^x \), when \( x = 0 \), \( y = 1 \), and as \( x \) increases, \( y \) increases rapidly. The top - left and bottom - left graphs show increasing behavior (rising as \( x \) increases), while the top - right and bottom - right show decreasing (decay) behavior. Now, check the \( y \) - intercept: for \( y = 6^x \), when \( x = 0 \), \( y=1 \), but let's check the value at \( x = 1 \): \( y = 6^1=6 \). Looking at the bottom - left graph: when \( x = 1 \), \( y\) seems to be around 6 (matches \( 6^1 = 6 \)). The top - left graph at \( x = 1 \) has \( y\) around, say, less than 6? Wait, no, let's re - evaluate. Wait, the function \( y = 6^x \): when \( x = 0 \), \( y = 1 \); when \( x = 1 \), \( y = 6 \); when \( x=- 1 \), \( y=\frac{1}{6}\approx0.17 \). The bottom - left graph: at \( x = 0 \), \( y\) is 1 (since it passes through \( (0,1) \) - like), and as \( x \) increases, it shoots up. Wait, actually, the bottom - left graph (third graph, if we count top - left as first, top - right as second, bottom - left as third, bottom - right as fourth) has a steeper increase, which matches \( y = 6^x \) (since \( b = 6 \) is a large base, so steep growth). But first, for the growth/decay part, we know it's growth.

Answer:

This function represents exponential \(\boldsymbol{\text{growth}}\). The graph of the function \(y = 6^x\) is the bottom - left graph (or the third graph in the set of four graphs, showing increasing behavior as \(x\) increases and passing through appropriate points for \(y = 6^x\)).