Which of the following is the graph of [tex]y=-2 \cdot 3^{5-x}[/tex]?

A. (Image of graph)
B. (Image of graph)
C. (Image of graph)

Rewrite the function: y = − 2 c d o t 3 5 − x = − 2 c d o t ( 3 1 ​ ) ( x − 5 ) .
Determine the horizontal asymptote: y = 0 .
Calculate the y-intercept: y = − 486 .
Identify the graph that is below the x-axis, increasing, has a horizontal asymptote at y = 0, and a y-intercept of -486: C ​ .

Explanation

Analyzing the Function We are given the function y = − 2 ⋅ 3 5 − x and asked to identify its graph. Let's analyze the function to understand its key features.

Rewriting the Function First, we can rewrite the exponent as 5 − x = − ( x − 5 ) . This tells us that the graph of y = 3 x is reflected about the y-axis and shifted 5 units to the right. The function becomes y = − 2 ⋅ 3 − ( x − 5 ) = − 2 ⋅ ( 3 − 1 ) ( x − 5 ) = − 2 ⋅ ( 3 1 ​ ) ( x − 5 ) .

Analyzing the Base The base of the exponential is 3 1 ​ , which is between 0 and 1, so the exponential part is a decreasing function. The negative sign in front reflects the graph about the x-axis, so the graph will be below the x-axis and increasing.

Finding the Horizontal Asymptote The horizontal asymptote is y = 0, since as x goes to infinity, 3 5 − x approaches 0.

Calculating the y-intercept To find the y-intercept, we set x = 0: y = − 2 ⋅ 3 5 − 0 = − 2 ⋅ 3 5 = − 2 ⋅ 243 = − 486 . So the y-intercept is -486.

Identifying the Graph Based on this analysis, we are looking for a graph that is below the x-axis, increasing, has a horizontal asymptote at y = 0, and a y-intercept of -486. Comparing these features with the given graphs, we can identify the correct one.


Examples
Exponential functions like y = − 2 c d o t 3 5 − x are used to model various phenomena in real life, such as radioactive decay, population growth, and compound interest. For example, if you invest money in an account that compounds interest continuously, the amount of money you have after a certain time can be modeled by an exponential function. Similarly, the decay of a radioactive substance can be modeled by an exponential function, where the amount of substance remaining decreases exponentially over time. Understanding the graphs of exponential functions helps us visualize and analyze these phenomena.

Answered by GinnyAnswer | 2025-07-08

The function y = − 2 ⋅ 3 5 − x represents a decreasing exponential graph that is reflected over the x-axis. It has a horizontal asymptote at y = 0 and a y-intercept of − 486 . The correct graph that fits these characteristics is option C.
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Answered by Anonymous | 2025-07-11