Question

chapter 1: exponential functions > section exercises 1.6 > exercise 10
tell whether the function $y = 7^x$ represents exponential growth or exponential decay
this function represents exponential
identify the graph of the function.
(there are four graphs shown, two with decaying curves and two with growing curves, on coordinate grids with x and y axes.)

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( y = a \cdot b^x \), where \( a>0 \) and \( b>0 \), \( b
eq1 \). If \( b > 1 \), it's exponential growth; if \( 0 < b < 1 \), it's exponential decay.
For \( y = 7^x \), here \( a = 1 \) and \( b = 7 \). Since \( 7>1 \), this is exponential growth.

Step2: Analyze the graph of \( y = 7^x \)

• When \( x = 0 \), \( y = 7^0=1 \)? Wait, no, wait: Wait, \( 7^0 = 1 \)? Wait, no, wait the function is \( y = 7^x \). Wait, when \( x = 0 \), \( y = 1 \)? Wait, but looking at the graphs, let's check the y - intercept. Wait, maybe I made a mistake. Wait, no, \( 7^0=1 \)? Wait, no, wait the standard exponential function \( y = b^x \) has y - intercept at \( (0,1) \) when \( a = 1 \). But in the graphs, some have y - intercept at 1? Wait, no, the graphs shown: let's see. Wait, the function \( y = 7^x \) is an exponential growth function, so as \( x \) increases, \( y \) increases rapidly, and as \( x \) decreases (goes to negative infinity), \( y \) approaches 0. Also, when \( x = 0 \), \( y = 1 \)? Wait, no, \( 7^0 = 1 \). Wait, but in the graphs, let's check the options. The exponential growth graphs are the ones that increase from left to right (as \( x \) increases, \( y \) increases) and have a y - intercept. Wait, the general shape of \( y = b^x \) with \( b>1 \) is increasing, passing through \( (0,1) \), and approaching 0 as \( x

ightarrow-\infty \). So among the given graphs, the ones that show increasing (going up as \( x \) increases) and have the correct y - intercept (around 1? Wait, maybe the graphs are scaled? Wait, no, let's re - evaluate. Wait, the function is \( y = 7^x \). Let's compute some points: when \( x = 1 \), \( y = 7 \); when \( x=-1 \), \( y=\frac{1}{7}\approx0.14 \). So the graph should pass through \( (0,1) \), increase rapidly as \( x \) increases, and approach 0 as \( x
ightarrow-\infty \). So the graph that is increasing (from left to right, going up) and has the correct behavior (approaching 0 as \( x \) is negative, increasing as \( x \) is positive) is the one representing exponential growth. So the function \( y = 7^x \) is exponential growth, and its graph is an increasing exponential curve, passing through \( (0,1) \) (or near that, depending on the graph scaling) and increasing as \( x \) increases.

Answer:

This function represents exponential growth. For the graph, the graph that shows an exponential growth (increasing as \( x \) increases, approaching 0 as \( x
ightarrow-\infty \)) is the correct one (e.g., if we consider the standard exponential growth graph shape, the one with the curve rising from the left - hand side (near the x - axis for negative \( x \)) and increasing steeply as \( x \) becomes positive).