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Chapter 1: Problem 14

Write a formula for \(f(x)=\frac{1}{x^{2}}\) shifted up 2 units and left 4 units.

Short Answer

Expert verified
The formula is \(f(x) = \frac{1}{(x+4)^2} + 2\).

Step by step solution

01

Understand the Base Function

The base function given here is \(f(x) = \frac{1}{x^2}\). This is a rational function and its graph is a hyperbola.
02

Horizontal Shift

To shift the function \(f(x)\) to the left by 4 units, substitute \(x+4\) for \(x\) in the function. Thus, the function becomes \(f(x) = \frac{1}{(x+4)^2}\).
03

Vertical Shift

To shift the function up by 2 units, add 2 to the current function \(\frac{1}{(x+4)^2}\). This leads to the new function \(f(x) = \frac{1}{(x+4)^2} + 2\).
04

Write the Final Function

After shifting left 4 units and up 2 units, the final expression for the function is \(f(x) = \frac{1}{(x+4)^2} + 2\).

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Key Concepts

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

Horizontal Shift
A horizontal shift in the graph of a function occurs when the graph is moved left or right. With the rational function \(f(x) = \frac{1}{x^2}\), a shift can be accomplished by replacing \(x\) with \(x \pm h\). To shift to the left, replace \(x\) with \(x + h\), where \(h\) is the number of units. For a shift to the right, replace \(x\) with \(x - h\).

In our problem, the graph is moved to the left by 4 units. This requires substituting \(x + 4\) for \(x\). Thus, \(f(x)\) becomes \(\frac{1}{(x+4)^2}\).

This transformation affects the graph:
  • Moves every point on the graph 4 units to the left on the x-axis.
  • The vertical asymptote at \(x = 0\) moves to \(x = -4\).
Understanding horizontal shifts will enable you to manipulate functions to align with desired points along the x-axis.
Vertical Shift
Vertical shifts involve moving a graph up or down on the coordinate plane. For the function \(f(x) = \frac{1}{x^2}\), a vertical shift is done by adding or subtracting a constant \(k\) to the function. If \(k\) is positive, the graph moves upward, and if negative, it moves downward.

In our example, we needed to shift the graph up by 2 units. This is achieved by adding 2 to the function \(\frac{1}{(x+4)^2}\), resulting in the transformed function \(f(x) = \frac{1}{(x+4)^2} + 2\).

Effects of the vertical shift include:
  • Each point on the graph moves 2 units higher on the y-axis.
  • The horizontal asymptote shifts from \(y = 0\) to \(y = 2\).
Being proficient with vertical shifts helps in adjusting the position of functions to meet specific vertical criteria or outcomes.
Rational Functions
Rational functions are functions that consist of ratios of polynomials. For example, \(f(x) = \frac{1}{x^2}\) is a rational function with a single polynomial in the denominator. Graphically, rational functions can exhibit hyperbolas, areas of undefined values (asymptotes), and interesting slopes.

Key features of rational functions include:
  • Vertical asymptotes, which occur where the denominator is zero, causing the function to be undefined.
  • Horizontal asymptotes, which describe the behavior of the graph as \(x\) approaches infinity or negative infinity.
  • Holes, which can appear where cancellations occur between the numerator and denominator.
Understanding rational functions allows students to predict function behavior in both the short and long term. Special attention to transformations such as horizontal and vertical shifts ensures clarity in altering these graphs for different mathematical and real-world applications.

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Most popular questions from this chapter

Select all of the following tables which represent \(y\) as a function of \(x\). a. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-4 & -2 \\ \hline 3 & 2 \\ \hline 6 & 4 \\ \hline 9 & 7 \\ \hline 12 & 16 \\ \hline \end{array} $$ b. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-5 & -3 \\ \hline 2 & 1 \\ \hline 2 & 4 \\ \hline 7 & 9 \\ \hline 11 & 10 \\ \hline \end{array} $$ c. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-1 & -3 \\ \hline 1 & 2 \\ \hline 5 & 4 \\ \hline 9 & 8 \\ \hline 1 & 2 \\ \hline \end{array} $$ d. $$ \begin{array}{|l|l|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline-1 & -5 \\ \hline 3 & 1 \\ \hline 5 & 1 \\ \hline 8 & 7 \\ \hline 14 & 12 \\ \hline \end{array} $$

Determine the interval(s) on which the function is concave up and concave down. $$ m(x)=-2(x+3)^{3}+1 $$

Write a formula for the function that results when the given toolkit function is transformed as described. \(f(x)=x^{2}\) horizontally compressed by a factor of \(\frac{1}{2},\) then shifted to the right 5 units and up 1 unit.

The amount of garbage, \(G,\) produced by a city with population \(p\) is given by \(G=f(p) . G\) is measured in tons per week, and \(p\) is measured in thousands of people. a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function \(f\). b. Explain the meaning of the statement \(f(5)=2\).

Use a graph to estimate the local extrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down. $$ f(x)=x^{4}-4 x^{3}+5 $$
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