Comparing Graphs Use a graphing utility to graph f(x)=e^x...

Name: Problem 130, Chapter 5: Calculus of a Single Variable
Uploaded: 2019-06-23T05:10:50-08:00
Duration: 48 s
Channel: Fuzail Shakir
Description: Comparing Graphs Use a graphing utility to graph f(x)=e^x and the given function in the same viewing window. How are the two graphs related? (a) g(x)=e^x-2 (b)h(x)=-(1)/(2) e^x (c) q(x)=e^-x+3

Comparing Graphs Use a graphing utility to graph $f(x)=e^{x}$ and the given function in the same viewing window. How are the two graphs related?
$$(a) g(x)=e^{x-2}$$
$$(b)h(x)=-\frac{1}{2} e^{x}$$
$$(c) q(x)=e^{-x}+3$$

Key Concepts

Vertical Translations

Vertical translations involve adding or subtracting a constant to the whole function, resulting in the graph shifting upward or downward respectively. This transformation does not change the shape of the graph but adjusts its vertical position on the coordinate plane.

Horizontal Reflection

A horizontal reflection is realized by replacing the variable x with -x in the function, which reflects the graph over the y-axis. This transformation changes the direction in which the function increases or decreases without altering other properties.

Vertical Scaling and Reflection

Vertical scaling is achieved by multiplying the function by a constant factor. If the constant is greater than 1, the graph stretches vertically; if it is between 0 and 1, the graph compresses. Additionally, if the constant is negative, the graph undergoes a reflection across the x-axis, flipping it upside-down.

Graphical Transformations

Graphical transformations are operations that modify the appearance and position of a function’s graph. These transformations include translations (shifting the graph horizontally or vertically), reflections (flipping the graph over an axis), and scalings (stretching or compressing the graph), and they are used to understand how changes to a function's formula affect its graph.

Exponential Functions

Exponential functions, such as f(x) = e^x, are functions where the variable appears in the exponent. They exhibit rapid growth or decay and have unique characteristics like a horizontal asymptote, a non-negative range (or patterned range after transformations), and a general shape that is influenced by any modifications made to the function’s formula.

Horizontal Shifts

A horizontal shift occurs when the input variable x is replaced by (x - h) in the function. This transformation moves the graph h units to the right (if h is positive) or to the left (if h is negative) without altering the shape of the original function.

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