Question

in the following graph, which represents the following exponential equation?
y = -2^x
options: b, e, none of these, c, d, a

Explanation:

Step1: Analyze the function \( y = -2^x \)

The parent function is \( y = 2^x \), which is an exponential growth function (since the base \( 2>1 \)) with a horizontal asymptote at \( y = 0 \), passing through \( (0,1) \) (when \( x = 0 \), \( y = 2^0=1 \)) and increasing as \( x \) increases. When we have \( y=-2^x \), this is a reflection of \( y = 2^x \) over the \( x \)-axis. So the key features of \( y=-2^x \) are:

• Horizontal asymptote at \( y = 0 \) (same as the parent, but reflected).
• When \( x = 0 \), \( y=-2^0=-1 \), so the graph passes through \( (0, - 1) \).
• As \( x \) increases, \( 2^x \) increases, so \( -2^x \) decreases (since the negative sign flips the direction), so it's a decreasing exponential function (approaching \( y = 0 \) from below as \( x\to+\infty \) and going to \( -\infty \) as \( x\to-\infty \), because when \( x\) is negative, \( 2^x=\frac{1}{2^{|x|}} \), so \( -2^x=-\frac{1}{2^{|x|}} \), which approaches \( 0 \) from below as \( x\to-\infty \)).

Step2: Compare with the given graph options

Looking at the graph (even with partial visibility), we need to identify the curve that:

• Passes through \( (0, - 1) \).
• Is decreasing (since as \( x \) increases, \( y=-2^x \) decreases).
• Has a horizontal asymptote at \( y = 0 \) (approaching from below).

Among the options, the curve that matches these features (decreasing, passing through \( (0, - 1) \), asymptote \( y = 0 \)) would be the one corresponding to \( y=-2^x \). From the typical graph of exponential functions, the reflection over the \( x \)-axis of \( y = 2^x \) (which is \( y=-2^x \)) will be a decreasing exponential curve passing through \( (0, - 1) \). Assuming the options have such a curve (like option D or similar, but from the analysis, we can conclude the correct graph is the one with these features).

Answer:

The graph representing \( y = -2^x \) is the one that is a decreasing exponential curve passing through \( (0, - 1) \) with horizontal asymptote \( y = 0 \) (the specific option, e.g., if the options are labeled, the one with these features; from the context, likely the curve that is decreasing and passes through \( (0, -1) \)).