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Chapter 3: Problem 29

Use the graph of \(f\) to describe the transformation that yields the graph of \(g .\) $$f(x)=\left(\frac{7}{2}\right)^{x}, \quad g(x)=-\left(\frac{7}{2}\right)^{-x}$$

Short Answer

Expert verified
To transform the graph of \(f(x) = \left(\frac{7}{2}\right)^x\) into the graph of \(g(x) = -\left(\frac{7}{2}\right)^{-x}\), we need to reflect \(f(x)\) around the x-axis, and then reflect the resulting graph around the y-axis. This means that any point (a, b) on \(f(x)\) corresponds to the point (-a, -b) on \(g(x)\).

Step by step solution

01

Identify the primary transaction: reflection around the x-axis

Observe that \(g(x)\) differs from \(f(x)\) by a negative sign in front. In terms of graphical transformations, this means that \(g(x)\) is the reflection of \(f(x)\) in the x-axis. If a point on \(f(x)\) has coordinates (a, b), its corresponding point on \(g(x)\) will be (a, -b).
02

Identify the secondary transformation: reflection around the y-axis

Next, notice that the exponent in \(g(x)\) is -x, while in \(f(x)\), the exponent is x. A negative exponent reflects the function around the y-axis. If a point on \(f(x)\) is at coordinates (a, b), the corresponding point on the graph of \(g(x)\) will be at (-a, b).
03

Combine the transformations

To obtain \(g(x)\) from \(f(x)\), the graph of \(f(x)\) is first reflected around the x-axis (due to the negative sign) and then reflected around the y-axis (due to the negative exponent). If a point on \(f(x)\) has coordinates (a, b), its corresponding point on \(g(x)\) will be (-a, -b) after the transformations.

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