Problem 5 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The x-intercept is \((1, 0)\) and the y-intercept is \((0, -\frac{1}{4})\).

Step 1: Finding the x-intercept

The x-intercept of a function is found where the output of the function is zero (i.e., where the numerator is zero). Solve for \(x\) in the equation \(x-1=0\). This gives \(x=1\).

Step 2: Finding the y-intercept

The y-intercept occurs where the input, \(x\), is zero. Substitute \(x = 0\) into the function: \[r(0) = \frac{0-1}{0+4} = -\frac{1}{4}.\] Thus, the y-intercept is \((0, -\frac{1}{4})\).

Key Concepts

Understanding the x-intercept

The x-intercept of a rational function is where the graph crosses the x-axis. This point occurs when the output of the function, denoted as \(r(x)\), equals zero. To find this x-intercept, focus on the numerator of the rational function. Why the numerator? Because a fraction only equals zero when its numerator is zero, assuming the denominator is not equal to zero at the same time.

Given the rational function \(r(x) = \frac{x-1}{x+4}\), you determine the x-intercept by setting the numerator \(x-1\) to zero and solving for \(x\):

• Set the equation: \(x-1=0\)
• Solve for \(x\): \(x=1\)

Thus, the x-intercept is \((1, 0)\). It's that simple! Remember, the x-intercept involves only the numerator because it decides when the entire function equals zero.

Determining the y-intercept

To find the point where the graph crosses the y-axis, known as the y-intercept, you evaluate the function at the input value of zero. This is straightforward: substitute \(x=0\) into the function. Why zero? Because the y-axis represents all points where \(x\) is zero.

Using \(r(x) = \frac{x-1}{x+4}\), substitute zero for \(x\):

• Compute: \(r(0) = \frac{0-1}{0+4} = -\frac{1}{4}\)

So, the y-intercept is \((0, -\frac{1}{4})\). This gives you the exact point on the graph where the line meets the y-axis. Evaluating at zero makes this process reliable and simple.

Exploring Numerator and Denominator

In rational functions, understanding both the numerator and the denominator is crucial. Each of these components has its own role in shaping the graph and finding intercepts.

• The **numerator** is responsible for determining the x-intercepts, as it must be zero for the function to zero out.
• The **denominator** influences the vertical asymptotes. These occur when the denominator is zero, creating undefined points in the function and potential breaks in the graph.

For \(r(x) = \frac{x-1}{x+4}\):

• The numerator \(x-1\) gives us the x-intercepts.
• The denominator \(x+4\) is considered when exploring vertical asymptotes, but it does not affect the intercepts directly.

This duality between numerator and denominator is what helps a rational function balance its behavior across different values of \(x\).

Methods for Finding Intercepts

Finding intercepts is a fundamental skill in algebra and calculus, especially for graph comprehension. The method for locating these intercepts is straightforward, involving distinct but related steps for x and y.

### Steps for Finding Intercepts:

• **X-Intercepts**: Set the numerator equal to zero and solve for \(x\). This pinpoints where the graph crosses the x-axis.
• **Y-Intercepts**: Substitute \(x=0\) into the function and solve for \(y\). This helps you understand where the graph touches the y-axis.

These steps are rooted in the function's structure and are applicable to any rational function you encounter. Standardizing this approach ensures accuracy and efficiency in graph analysis. By grasping these concepts, you're well on your way to mastering rational functions and their graphical behavior.