24.07.2022 - 20:22

# Make a rough sketch of the graph of the function , y = 2(1-e^x)., . Do not use a calculator. Just use an existing graph of , y=e^x, and, if necessary, the transformations of translation, stretching/shrinking, or reflecting.

Question:

Make a rough sketch of the graph of the function {eq}, y = 2(1-e^x)., {/eq}. Do not use a calculator. Just use an existing graph of {eq}, y=e^x, {/eq} and, if necessary, the transformations of translation, stretching/shrinking, or reflecting.

Answers (1)
• April 12, 2023 в 04:41
To sketch the graph of the function {eq}y = 2(1-e^x){/eq}, we can start with the graph of {eq}y = e^x{/eq}, which is an exponential function that passes through the point (0, 1) and has a vertical asymptote at {eq}x = -infty{/eq} and a horizontal asymptote at {eq}y = 0{/eq}. Next, we can apply the transformations to this graph to obtain the graph of {eq}y = 2(1-e^x){/eq}. The first transformation we apply is a reflection about the x-axis, which changes the sign of the y-values. This gives us the graph of {eq}y = -e^x{/eq}, which is the same as {eq}y = e^{-x}{/eq}. The next transformation we apply is a vertical stretching/shrinking by a factor of 2, which doubles the distance between the horizontal asymptote and the graph. This gives us the graph of {eq}y = 2e^{-x}{/eq}. Finally, we translate the graph upwards by 2 units, which moves the horizontal asymptote from {eq}y = 0{/eq} to {eq}y = 2{/eq}, and moves the point (0, 1) to (0, 3). This gives us the graph of {eq}y = 2(1-e^x){/eq}. Overall, the graph of {eq}y = 2(1-e^x){/eq} is a decreasing exponential function that starts at (0, 2), approaches the horizontal asymptote at {eq}y = 2{/eq} as {eq}x to +infty{/eq}, and approaches the x-axis as {eq}x to -infty{/eq}.
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