The two images you provided relate to an exponential function $f(x) = 2 \left(\frac{1}{2}\right)^x - 5$, and the task asks for identifying its features. Let's break it down based on the graph and the information on the screen:

Function Analysis

The function is: $f(x) = 2 \left(\frac{1}{2}\right)^x - 5$

This is an exponential decay function because the base of the exponent, $\frac{1}{2}$, is between 0 and 1.

Key Features:

1. Type of function: This is an exponential function.
2. Asymptote: There is a horizontal asymptote at $y = -5$. This is due to the "-5" added to the exponential part, which shifts the graph downwards by 5 units.
3. Range: The range of the function is $(-5, \infty)$, meaning it never touches or goes below $y = -5$.
4. Domain: The domain is all real numbers, i.e., $(-\infty, \infty)$, as exponential functions are defined for all $x$.
5. End Behavior:
   • As $x \to -\infty$, $y \to \infty$.
   • As $x \to \infty$, $y \to -5$, because the exponential decays towards 0, but the shift by -5 causes it to approach $y = -5$.

Filling the Gaps in the Answer

Using the graph and the features, here’s how the answer should be filled in:

• The function $f(x)$ is an exponential function with a horizontal asymptote of $y = -5$.
• The range of the function is $(-5, \infty)$, and it is decreasing on its domain of $(-\infty, \infty)$.
• The end behavior on the LEFT side is as $x \to -\infty$, $y \to \infty$.
• The end behavior on the RIGHT side is as $x \to \infty$, $y \to -5$.

Would you like any further clarification on these features?

Questions to Consider:

1. What effect does changing the base of the exponent have on the graph?
2. How does the vertical shift (-5 in this case) influence the function's range and asymptote?
3. Why does the function decrease instead of increase as $x \to \infty$?
4. How would the graph change if the base was greater than 1 instead of $\frac{1}{2}$?
5. What is the significance of the horizontal asymptote in real-world contexts?

Tip:

Exponential decay functions have bases between 0 and 1, causing them to decrease as $x$ increases, while the horizontal asymptote determines the lowest value the function approaches but never reaches.