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Chapter 3: Problem 5

Find the \(x\)- and \(y\)-intercepts of the rational function. $$r(x)=\frac{x-1}{x+4}$$

Short Answer

Expert verified
The x-intercept is (1, 0) and the y-intercept is (0, -1/4).

Step by step solution

01

Identify the x-intercept

The x-intercept of a function is the point where the graph crosses the x-axis. For rational functions, this occurs where the numerator is zero, because that is the point where the output value is zero. Set the numerator equal to zero and solve for x:\[x - 1 = 0 \]Solving this equation, we find that \(x = 1\). Therefore, the x-intercept is at the point \((1, 0)\).
02

Find the y-intercept

The y-intercept of a function is the point where the graph crosses the y-axis, which is where \(x = 0\). Substitute \(x = 0\) in the function:\[r(0) = \frac{0 - 1}{0 + 4} = \frac{-1}{4} \]Thus, the y-intercept is at the point \((0, -\frac{1}{4})\).

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Key Concepts

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

x-intercepts
Finding the x-intercepts of a rational function is crucial for understanding where the function meets the x-axis. These intercepts occur where the function's value is zero, which happens when the numerator equals zero. For the function given, \(r(x) = \frac{x-1}{x+4}\), we start by setting the numerator \(x-1\) to zero:
  • Set \(x - 1 = 0\)
  • Solve for \(x\): \(x = 1\)
This tells us that the function intersects the x-axis at \((1, 0)\). In simpler terms, when \(x = 1\), the value of the entire function becomes zero, showing the point of intersection on the x-axis.
y-intercepts
Identifying the y-intercept of a rational function involves finding the point at which the function crosses the y-axis. For any function, this happens when \(x = 0\). We substitute \(x = 0\) into our function \(r(x) = \frac{x-1}{x+4}\):
  • Substitute: \(r(0) = \frac{0-1}{0+4} = \frac{-1}{4}\)
This calculation shows that the y-intercept is at the point \((0, -\frac{1}{4})\). Here, the y-value of \(-\frac{1}{4}\) represents where the function meets the y-axis, illustrating where the graph starts from the y-axis when followed from the origin.
numerator of a rational function
The numerator of a rational function plays a key role in determining both the x-intercepts and the behavior of the graph. In the rational function \(r(x) = \frac{x-1}{x+4}\), the numerator is \(x-1\). The x-intercepts, specifically, occur when this numerator equals zero, because it leads the entire expression to become zero.
  • Reason for x-intercepts: \(x - 1 = 0\) implies \(x = 1\)
  • Significance: Whenever the numerator is zero, the output of the function is zero.
Understanding the numerator is essential because it directly influences where the graph of the function will touch or cross the x-axis.
graph of a function
Creating a graph of a rational function helps visualize its behavior and key characteristics like intercepts and asymptotes. For \(r(x) = \frac{x-1}{x+4}\), the graph is a helpful tool for identifying its x-intercept at \((1, 0)\) and y-intercept at \((0, -\frac{1}{4})\). Rational functions can have vertical asymptotes, which correspond to the values that make the denominator zero. For this function, \(x = -4\) makes the denominator zero, indicating an asymptote.
  • X-intercept: Occurs at \((1, 0)\).
  • Y-intercept: Occurs at \((0, -\frac{1}{4})\).
  • Vertical Asymptote: Occurs at \(x = -4\).
Graphing the function, keeping these intercepts and asymptotes in mind, reveals how the function behaves around these critical points, helping provide a fuller picture of its overall form.

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Most popular questions from this chapter

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1}$$

Show that the polynomial does not have any rational zeros. $$P(x)=x^{50}-5 x^{25}+x^{2}-1$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{3 x-x^{2}}{2 x-2}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{4 x^{2}}{x^{2}-2 x-3}$$

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}}{x^{2}-2}$$
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