Problem 22 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$g(x)=\frac{1}{x+2}+2$$

Step-by-Step Solution

Answer

The function \( g(x)=\frac{1}{x+2}+2 \) has a y-intercept at \( g(0)=2.5 \). The vertical asymptote is \( x=-2 \), while the horizontal asymptote is \( g(x) = 2 \). The function has no holes. The graph of the function has been sketched taking into account all the characteristics mentioned and compared with a graphing utility for confirmation.

Step 1: Determine the intercepts

The x-intercept is determined by setting \( g(x) = 0 \), which yields no solution in this case. The y-intercept is found by evaluating the function at \( x=0 \), giving \( g(0)=\frac{1}{0+2}+2=2.5 \)

Step 2: Determine the vertical asymptotes

Vertical asymptotes are found by evaluating the denominator of the rational function. The function becomes undefined when the denominator equals 0. Therefore, set \( x+2 = 0 \), implying \( x = -2 \), which is the equation of the vertical asymptote.

Step 3: Determine the horizontal asymptotes

A horizontal asymptote is found by evaluating the limit of the function as \( x \) approaches positive or negative infinity. Since the polynomial in the denominator has a greater degree than that in the numerator, both \( \lim_{x\to\infty} g(x) \) and \( \lim_{x\to -\infty} g(x) \) yield 2, which is the equation of the horizontal asymptote.

Step 4: Verify using a graphing utility

With the intercepts, asymptotes and their behaviour at each side, now make a sketch of the function. The graph should accurately reflect all these details. To be sure if this is a perfect sketch, compare it with a graph from a graphing utility.

Step 5: Determine possible holes

Holes, or excluded values, on a rational function's graph are values for which both the numerator and the denominator are zero. Here, the denominator is zero only for \( x=-2 \), but the numerator does not equal zero for that value. Hence, the function has no holes.

Key Concepts

Rational Function Intercepts

Understanding intercepts is crucial when graphing a rational function like \( g(x)=\frac{1}{x+2}+2 \). \
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Intercepts are the points where the graph crosses the axes. To find the y-intercept, we simply input \( x=0 \) into the function and calculate \( g(0) \). In our case, \( g(0)=\frac{1}{0+2}+2=2.5 \), which means the graph crosses the y-axis at (0, 2.5). \
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X-Intercept of the Rational Function\

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For the x-intercept, we need to find the value of \( x \) that will make \( g(x)=0 \). If no real solution exists, like with our function \( g(x) \) that never hits the x-axis, then this function has no x-intercept.

Vertical Asymptotes

Vertical asymptotes are vertical lines where the graph of a function approaches but never touches. They occur in places where the function is undefined—typically where the denominator of a rational function is zero. \
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To determine vertical asymptotes for \( g(x)=\frac{1}{x+2}+2 \), we solve \( x+2=0 \), leading to \( x=-2 \). Hence, \( x=-2 \) is our vertical asymptote. The function blows up to infinity or negative infinity as \( x \) nears -2, but never reaches -2 itself.

Horizontal Asymptotes

Horizontal asymptotes represent the value that a function's output approaches as \( x \) goes to infinity or negative infinity. They're determined by the end behavior of the function. \
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For \( g(x)=\frac{1}{x+2}+2 \), we compare the degrees of the numerator and the denominator. Since the numerator has a lower degree, as \( x \) gets larger and larger, the \( \frac{1}{x+2} \) part of \( g(x) \) gets closer and closer to 0, and \( g(x) \) approaches 2. This means our function has a horizontal asymptote at \( y=2 \) which the graph will approach but not cross as \( x \) tends to positive or negative infinity.

Graph Sketching

Sketching the graph of a rational function involves combining the previously identified aspects—the intercepts, asymptotes, and general behavior of the function.\
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Plotting Key Components\

Start by plotting the y-intercept (0, 2.5). Next, draw a dotted vertical line at \( x=-2 \) to represent the vertical asymptote. The horizontal asymptote along \( y=2 \) should be represented by a dotted horizontal line. \
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Creating the Graph\

Knowing that the graph approaches the intercepts and the asymptotes, we can start plotting points to shape the graph. We also know that this graph does not intersect the x-axis and it lies above the horizontal asymptote once \( x \) is greater than -2 and below the asymptote when \( x \) is less than -2. With these points and the knowledge of asymptotes, you can sketch a curve that comes close to these lines but doesn't cross them, verifies these behaviors, and so completing the sketch of \( g(x) \).