Problem 11 - 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

x-intercept: (1, 0); y-intercept: \(\left(0, \frac{-1}{4}\right)\).

Step 1: Finding the y-intercept

To find the y-intercept of the function, we set \(x = 0\) in the function \(r(x) = \frac{x - 1}{x + 4}\). Substituting \(x = 0\), we get \(r(0) = \frac{0 - 1}{0 + 4} = \frac{-1}{4}\). Therefore, the y-intercept is \(\left(0, \frac{-1}{4}\right)\).

Step 2: Finding the x-intercept

To find the x-intercept of the function, we set the numerator of the function equal to zero. So, for \(r(x) = \frac{x - 1}{x + 4}\), we set \(x - 1 = 0\). This gives \(x = 1\). Therefore, the x-intercept is \((1, 0)\).

Key Concepts

Finding the Y-Intercept

When working with rational functions, one of the first tasks might be to identify the y-intercept. The y-intercept of a function is the point at which the graph of the function intersects the y-axis. To find this point, you need to set the variable \(x\) to zero in the function equation. Let's see how this works with our example rational function, \( r(x) = \frac{x - 1}{x + 4} \).

By plugging \(x = 0\) into the function, you simplify it to find \(r(0)\) which represents the y-intercept. This results in:

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

This means the point \( (0, \frac{-1}{4}) \) is where the function crosses the y-axis. Remember, intercepts can help you understand where the graph will intersect certain axes, helping you build a mental map of how the overall structure of a function looks.

Finding the X-Intercept

The x-intercept is equally important when analyzing rational functions. This is the point where the graph intersects the x-axis, meaning where the value of the function is zero. To find the x-intercept, set the numerator of the rational function equal to zero, since a fraction equals zero when its numerator is zero (and the denominator is non-zero).

For the function \( r(x) = \frac{x - 1}{x + 4} \), set the numerator \(x - 1\) to zero:

• \(x - 1 = 0\)
• Solving for \(x\) gives \(x = 1\)

This tells us that the x-intercept is at the point \((1, 0)\). Since the function is zero at this point, it directly strikes the x-axis. The x-intercept provides significant insights into the roots or solutions of the function. Finding intersections like these is essential in understanding how the function behaves on the graph.

Understanding Rational Functions

Rational functions are a central concept in mathematics, often appearing in calculus and college algebra. A rational function is essentially a ratio (or fraction) of two polynomials. It is expressed typically in the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials. This structure creates interesting characteristics, such as intercepts and vertical asymptotes.

Here's what to keep in mind:

• **Domains:** These functions often have restricted domains because the denominator \(Q(x)\) must not be zero.
• **Intersections:** Finding x- and y-intercepts help graph and interpret the function.
• **Behavior:** Rational functions can have horizontal or oblique asymptotes, which describe the function’s end-behavior.

Understanding these elements can transform the analysis of a rational function, providing a fuller picture of what the graph looks like and how the function behaves in different scenarios. Keep practicing these steps, and you will master working with rational functions and easily identify their critical features.