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Chapter 8: Problem 9

For each function, construct a new function whose graph is the graph of the original function shifted left by two units, then multiplied by \(\frac{1}{3}\), and then shifted down by five units. a. \(f(x)=60\left(\frac{1}{2}\right)^{x}\) b. \(g(x)=12 x^{3}\) c. \(y=\log x\)

Short Answer

Expert verified
a. \g(x) = 20\left(\frac{1}{2}\right)^{x+2} - 5\b. \h(x) = 4(x+2)^3 - 5\c. \k(x) = \frac{1}{3} \log(x+2) - 5\

Step by step solution

01

Shift left by two units

To shift the graph of a function left by two units, replace every occurrence of x in the original function with \(x + 2\).
02

Multiply by \(\frac{1}{3}\)

To multiply the graph of a function by \(\frac{1}{3}\), multiply the entire function by \(\frac{1}{3}\).
03

Shift down by five units

To shift the graph of a function down by five units, subtract 5 from the entire function.
04

Apply transformations to function \(f(x)=60\left(\frac{1}{2}\right)^x\)

First, shift left by two units: \(f(x + 2) = 60\left(\frac{1}{2}\right)^{x+2}\). Then, multiply by \(\frac{1}{3}\): \(\frac{1}{3} \cdot 60\left(\frac{1}{2}\right)^{x+2}\). Finally, shift down by five units: \(\frac{1}{3} \cdot 60\left(\frac{1}{2}\right)^{x+2} - 5\). Simplified, this is \g(x) = 20\left(\frac{1}{2}\right)^{x+2} - 5\.
05

Apply transformations to function \(g(x)=12 x^3\)

First, shift left by two units: \(g(x + 2) = 12(x+2)^3\). Then, multiply by \(\frac{1}{3}\): \(\frac{1}{3} \cdot 12(x+2)^3\). Finally, shift down by five units: \(\frac{1}{3} \cdot 12(x+2)^3 - 5\). Simplified, this is \h(x) = 4(x+2)^3 - 5\.
06

Apply transformations to function \(y=\log x\)

First, shift left by two units: \log(x+2)\. Then, multiply by \(\frac{1}{3}\): \(\frac{1}{3} \log(x+2)\). Finally, shift down by five units: \(\frac{1}{3} \log(x+2) - 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graph shifting
Graph shifting is a fundamental transformation that moves the entire graph of a function either horizontally or vertically. To shift a graph to the left by two units, replace every occurrence of x in the function with \(x + 2\). This shifts every point on the graph two units to the left without altering the function's shape. For example, in the function \(f(x) = 60(\frac{1}{2})^x\), transforming it to \(f(x + 2) = 60(\frac{1}{2})^{x+2}\) moves its graph two units left.
function multiplication
Function multiplication involves multiplying the entire function by a constant. This transformation changes the function's vertical scale. For instance, multiplying the function by \(\frac{1}{3}\) scales the y-values of the graph, making it narrower or wider depending on the constant. For example, to multiply \(60(\frac{1}{2})^{x+2}\) by \(\frac{1}{3}\), the resultant function is \(20(\frac{1}{2})^{x+2}\). This scaling transformation can make the graph appear stretched or compressed vertically.
logarithmic functions
Logarithmic functions are a type of function where the variable is inside a logarithm. For example, \(y = \log x\) is a logarithmic function. Applying transformations to logarithmic functions follow the same rules as other functions. To shift \(y = \log x\) left by two units and transform it, replace \(x\) by \(x + 2\) to get \(y = \log (x + 2)\). Multiplying by \(\frac{1}{3}\), yields \(y = \frac{1}{3} \log (x + 2)\). Lastly, shifting it down by five units gives \(y = \frac{1}{3} \log (x + 2) - 5\).
polynomial functions
Polynomial functions involve variables raised to positive integer powers. For example, \(g(x)=12x^3\) is a polynomial function. Shifting it left by two units and transforming it means replacing \(x\) by \(x + 2\), resulting in \(g(x + 2) = 12(x+2)^3\). Then, multiplying by \(\frac{1}{3}\) leads to \(4(x+2)^3\). Finally, shifting it down by five units results in \(g(x) = 4(x+2)^3 - 5\).

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Most popular questions from this chapter

(Graphing program required for part (c).) The rational function \(g(x)=\frac{4 x-11}{x-3}\) can be decomposed into a sum by using the following method: a. Use the preceding method to decompose \(g(x)=\frac{5 x+22}{x-3}\). b. Describe the transformation of the function \(f(x)=\frac{1}{x}\) into \(g(x)=\frac{5 x+22}{x-3}\) c. Using technology, plot the graphs of \(f(x)\) and \(g(x)\) to verify that the transformation described in part (b) is correct.

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A stone is dropped into a pond, causing a circular ripple that is expanding at a rate of \(13 \mathrm{ft} / \mathrm{sec}\). Describe the area of the circle as a function of time.
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