Question

quiz: exponential functions hs: algebra 2b m (sequential) / 3:exponentials a (image of a graph with x-axis from -4 to 4 and y-axis from -4 to 4, showing an exponential curve) options: y = -1, y = -2, y = -3, y = 0

Explanation:

Step 1: Recall the concept of horizontal asymptote for exponential functions

For an exponential function of the form \( y = a\cdot b^{x}+k\) (where \( a
eq0\), \( b>0\), \( b
eq1\)), the horizontal asymptote is the line \( y = k\). This is because as \( x\to\pm\infty\) (depending on the base \( b\)), the term \( a\cdot b^{x}\) will either approach \( 0\) (if \( 0 < b<1\) and \( x\to+\infty\), or if \( b > 1\) and \( x\to-\infty\)) and the function will approach \( y = k\).

Step 2: Analyze the given exponential function graph

Looking at the provided graph of the exponential function, as \( x\) approaches negative infinity (\( x\to-\infty\)), the graph of the function gets closer and closer to a horizontal line. By observing the y - coordinates and the grid lines, we can see that this horizontal line (the horizontal asymptote) corresponds to \( y=-2\).

Answer:

(Assuming the question is to find the horizontal asymptote of the exponential function graph) The horizontal asymptote is \( y = - 2 \)? Wait, no, let's re - analyze. Wait, the graph of the exponential function: for an exponential function of the form \( y=a\cdot b^{x}+k \), the horizontal asymptote is \( y = k \). Looking at the graph, as \( x\to-\infty \), the function approaches \( y=-2 \)? Wait, no, the graph is approaching a horizontal line. Wait, the options are \( y = - 1\), \( y=-2\), \( y = - 3\), \( y = 0\). Wait, maybe I made a mistake. Wait, the graph: when \( x\) is very small (negative), the curve is approaching a horizontal line. Let's check the y - axis. The grid lines: the horizontal line that the left - end of the curve is approaching. Let's see the y - values. The curve, as \( x\to-\infty \), is getting closer to \( y=-2 \)? Wait, no, maybe the correct horizontal asymptote is \( y=-2 \)? Wait, no, let's look again. Wait, the graph: the lowest part, as \( x\) goes to negative infinity, the function approaches \( y = - 2\)? Wait, maybe the correct answer is \( y=-2 \), so the option is the one with \( y = - 2\). But wait, maybe I misread. Wait, the graph: let's see the y - axis. The horizontal asymptote of an exponential function \( y = a\cdot b^{x}+c \) is \( y = c \). From the graph, when \( x\) is very negative, the function is approaching \( y=-2 \)? Wait, no, maybe the correct answer is \( y=-2 \), so the option is \( y=-2 \) (the second option, with the circle next to \( y = - 2\)).

Wait, maybe I made a mistake. Let's re - examine. The graph: the curve is an exponential growth curve (since it's increasing as \( x\) increases). The horizontal asymptote is the line that the function approaches as \( x\to-\infty \). Looking at the graph, when \( x\) is very negative, the function is approaching \( y=-2 \). So the horizontal asymptote is \( y=-2 \), so the correct option is the one with \( y=-2 \) (the second radio button option: \( y = - 2\)).