Problem 7
Question
Applying what you learned in the last section of this chapter to the pocketful of functions you ve been introduced to (the identity, squaring, reciprocal, and absolute value functions), graph the following functions. Label any asymptotes and \(x\) - and \(y\) -intercepts. (a) \(f(x)=\frac{1}{x^{2}}\) (b) \(g(x)=\left|(x-1)^{2}\right|\) (c) \(h(x)=\left|x^{2}-1\right|\) (d) \(j(x)=\frac{1}{x+1}+2\) (e) \(m(x)=\frac{-1}{x-2}+1\) (f) \(p(x)=\left|\frac{1}{x}\right|\)
Step-by-Step Solution
Verified Answer
The graphs of the given functions show different properties such as shifts, asymptotes, and reflections. By analyzing the form of each function, these properties can be identified and that's how the functions can be graphed.
1Step 1: Graphing the Reciprocal Function
For the function \(f(x)=\frac{1}{x^{2}}\), it's a reciprocal function. When \(x=0\), the function is undefined, so there is a vertical asymptote at \(x=0\). Also, as \(x\) approaches infinity, \(f(x)\) approaches 0, so there is a horizontal asymptote at \(y=0\). The y-intercept is also undefined in this case.
2Step 2: Graphing the Absolute Square Function
The function \(g(x)=\left|(x-1)^{2}\right|\) is simply a square function shifted right one unit. It does not have any asymptotes and its x-intercept is \(x=1\), which is obtained by setting the function to zero and solving for \(x\).
3Step 3: Graphing the Absolute Value of a Quadratic Function
For the function \(h(x)=\left|x^{2}-1\right|\), it's the absolute value of a quadratic function. It has x-intercepts at \(x=-1\) and \(x=1\), and has no asymptotes.
4Step 4: Graphing the Reciprocal Function with a Shift and Vertical Shift
The function \(j(x)=\frac{1}{x+1}+2\) is the reciprocal function shifted left one unit and up two units. Its vertical asymptote is at \(x=-1\) and its horizontal asymptote at \(y=2\). It crosses the y-axis at \(y=3\).
5Step 5: Graphing the Reciprocal Function with a Shift and Reflection
For \(m(x)=\frac{-1}{x-2}+1\), it's essentially the reciprocal function shifted right two units, reflected across the x-axis, and shifted up one unit. Its vertical asymptote is at \(x=2\) and the horizontal asymptote at \(y=1\).
6Step 6: Graphing the Absolute Value of a Reciprocal Function
Finally, the function \(p(x)=\left|\frac{1}{x}\right|\) is the absolute value of the reciprocal function. It has a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\).
Key Concepts
Reciprocal FunctionAbsolute Value FunctionsAsymptotesX-InterceptsY-Intercepts
Reciprocal Function
Understanding reciprocal functions like \(f(x)=\frac{1}{x^{2}}\) is fundamental in calculus. A reciprocal function is simply one where \(y\) is inversely proportional to \(x\). Here's what happens: as \(x\) gets larger, \(f(x)\) gets smaller because they are inversely related. This relationship creates a hyperbola, a common graph shape for reciprocal functions.
There are two main features to look out for when graphing reciprocal functions: vertical and horizontal asymptotes. A vertical asymptote is a line that the graph approaches but never touches or crosses, indicating points where the function is undefined. For instance, \(f(x)=\frac{1}{x^{2}}\) has a vertical asymptote at \(x=0\) because division by zero is undefined. Horizontal asymptotes show the value that \(f(x)\) approaches as \(x\) goes to infinity or negative infinity. In the case of \(f(x)\), it has a horizontal asymptote at \(y=0\), meaning as \(x\) becomes very large, \(f(x)\) gets closer and closer to zero.
There are two main features to look out for when graphing reciprocal functions: vertical and horizontal asymptotes. A vertical asymptote is a line that the graph approaches but never touches or crosses, indicating points where the function is undefined. For instance, \(f(x)=\frac{1}{x^{2}}\) has a vertical asymptote at \(x=0\) because division by zero is undefined. Horizontal asymptotes show the value that \(f(x)\) approaches as \(x\) goes to infinity or negative infinity. In the case of \(f(x)\), it has a horizontal asymptote at \(y=0\), meaning as \(x\) becomes very large, \(f(x)\) gets closer and closer to zero.
Absolute Value Functions
Absolute value functions, like \(g(x)=|(x-1)^{2}|\) and \(h(x)=|x^{2}-1|\), involve taking the absolute value of a function. The absolute value ensures that all outputs are non-negative, which means the graph will only be in the first and second quadrants. These functions reflect any negative part of the graph above the \(x\)-axis.
Graphing these functions usually starts with graphing the inside function without the absolute value. Then, reflect the parts of the graph that lie below the \(x\)-axis onto the upper side. It's like having a mirror on the \(x\)-axis! Absolute value functions do not have asymptotes, as the outcomes stretch infinitely along the positive \(y\)-axis without approaching a specific value. Their x-intercepts are found where the function equals zero. With the given functions, the x-intercepts are easy to find with basic algebra.
Graphing these functions usually starts with graphing the inside function without the absolute value. Then, reflect the parts of the graph that lie below the \(x\)-axis onto the upper side. It's like having a mirror on the \(x\)-axis! Absolute value functions do not have asymptotes, as the outcomes stretch infinitely along the positive \(y\)-axis without approaching a specific value. Their x-intercepts are found where the function equals zero. With the given functions, the x-intercepts are easy to find with basic algebra.
Asymptotes
Asymptotes are lines that a graph approaches but never actually reaches. They're like cosmic boundaries—there, but untouchable. They can be vertical, horizontal, or even slanted. A vertical asymptote, as mentioned, occurs where the function cannot have a value, typically where there's a division by zero. Horizontal asymptotes indicate the value that \(y\) approaches as \(x\) tends toward infinity or negative infinity.
To find asymptotes, look for points where the function becomes undefined. For horizontal asymptotes of rational functions, compare the degrees of the polynomials in the numerator and the denominator. If the degree in the numerator is less, the horizontal asymptote is the \(x\)-axis, or \(y=0\). If they're equal, then the horizontal asymptote is the ratio of the leading coefficients. Asymptotes give us valuable clues about the function's behavior at the extremes and greatly aid in sketching the graph.
To find asymptotes, look for points where the function becomes undefined. For horizontal asymptotes of rational functions, compare the degrees of the polynomials in the numerator and the denominator. If the degree in the numerator is less, the horizontal asymptote is the \(x\)-axis, or \(y=0\). If they're equal, then the horizontal asymptote is the ratio of the leading coefficients. Asymptotes give us valuable clues about the function's behavior at the extremes and greatly aid in sketching the graph.
X-Intercepts
The x-intercepts of a function are the points where the graph crosses the \(x\)-axis. They're the solutions to the equation \(f(x)=0\). In other words, they tell us for which values of \(x\), the output of the function is zero. Finding x-intercepts is often one of the first steps in graphing because it gives us specific points through which the graph will pass.
For example, to find the x-intercepts of \(g(x)=|(x-1)^{2}|\), you'd set the function equal to zero and solve for \(x\), which would show us that the x-intercept is \(x=1\). With x-intercepts, we start piecing together the puzzle of what the function looks like across its domain - it is the first clue in unraveling the mystery of its graph.
For example, to find the x-intercepts of \(g(x)=|(x-1)^{2}|\), you'd set the function equal to zero and solve for \(x\), which would show us that the x-intercept is \(x=1\). With x-intercepts, we start piecing together the puzzle of what the function looks like across its domain - it is the first clue in unraveling the mystery of its graph.
Y-Intercepts
The y-intercept, on the other hand, is where the function crosses the y-axis. This point tells you what \(f(x)\) equals when \(x=0\). It's like the function's starting point if you're moving along the graph from left to right. To find the y-intercept, simply plug zero into the function for \(x\).
However, not all functions have y-intercepts. For instance, the graph of \(f(x)=\frac{1}{x^{2}}\) doesn't have a y-intercept, because the function is undefined at \(x=0\). In contrast, for \(j(x)=\frac{1}{x+1}+2\), plugging in zero gives us a y-intercept of 3. Y-intercepts complete the snapshot of where our function graph begins or crosses the vertical serial line, further helping in mapping out the journey of a function's graph on the plane.
However, not all functions have y-intercepts. For instance, the graph of \(f(x)=\frac{1}{x^{2}}\) doesn't have a y-intercept, because the function is undefined at \(x=0\). In contrast, for \(j(x)=\frac{1}{x+1}+2\), plugging in zero gives us a y-intercept of 3. Y-intercepts complete the snapshot of where our function graph begins or crosses the vertical serial line, further helping in mapping out the journey of a function's graph on the plane.
Other exercises in this chapter
Problem 3
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