7-12 Make a rough sketch of the graph of the function. Do...

$7-12$ Make a rough sketch of the graph of the function. Do not
use a calculator. Just use the graphs given in Figures 3 and 14
and, if necessary, the transformations of Section 1.3
$$y=2\left(1-e^{x}\right)$$

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Key Concepts

Exponential Functions

Exponential functions, such as y = e^x, are functions where the variable appears in the exponent. They exhibit rapid growth or decay, have a constant base, and are common in modeling natural phenomena like population growth and radioactive decay.

Function Transformations

Function transformations involve modifying a base function through operations such as translations (shifts), reflections, and scalings. They allow us to predict and sketch the behavior of complex functions by understanding how these operations alter the original graph.

Vertical Scaling

Vertical scaling refers to multiplying a function by a constant, which stretches or compresses the graph vertically. In the function y = 2(1 - e^x), the factor of 2 scales the transformed exponential function, affecting its overall amplitude.

Graph Reflection and Translation

Graph reflection involves flipping a function’s graph across an axis, while translation moves the graph without changing its shape. In y = 2(1 - e^x), the transformation inside the parentheses (1 - e^x) can be decomposed as a reflection of e^x followed by a translation, altering the position and direction of the asymptotes and key features of the graph.

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