Problem 16 - Algebra and Trigonometry

Question

Find the \(x\) -and \(y\) -intercepts of the rational function. $$ r(x)=\frac{x^{3}+8}{x^{2}+4} $$

Step-by-Step Solution

Answer

The x-intercept is at \((-2, 0)\) and the y-intercept is at \((0, 2)\).

Step 1: Find the x-intercept

To find the x-intercept(s), set the numerator of the function equal to zero, since the x-intercept of a fraction occurs where the numerator is zero (and the denominator is not zero). The numerator of the function is \(x^3 + 8\). Set it to zero:\[x^3 + 8 = 0\]Solve for \(x\):\[x^3 = -8\]\[x = -2\]The x-intercept is at \((x = -2, y = 0)\).

Step 2: Find the y-intercept

To find the y-intercept, evaluate the function where \(x = 0\) since the y-intercept happens when \(x = 0\). Substitute \(x=0\) into the function: \[r(0) = \frac{0^3 + 8}{0^2 + 4} = \frac{8}{4} = 2\]The y-intercept is at \((x = 0, y = 2)\).

Step 3: Verify the Denominator is Non-Zero at x-intercept

Double-check that the denominator is not zero at the x-intercept found in Step 1. At \(x = -2\), the denominator is: \[(-2)^2 + 4 = 4 + 4 = 8 \]Since 8 is not zero, the original x-intercept \((x = -2, y = 0)\) is valid.

Key Concepts

X-Intercepts

Understanding the concept of x-intercepts is essential in graphing and analyzing rational functions. An x-intercept of a function is a point where the graph crosses the x-axis. At this point, the value of the function is zero (i.e., the output or y-coordinate is zero). To find the x-intercept of a rational function like \( r(x)=\frac{x^{3}+8}{x^{2}+4} \), follow these steps:

• Focus on the numerator. This is because the output of the function equals zero when the numerator is zero (and the denominator is non-zero).
• Write the equation of the numerator equal to zero. For \( r(x) \), this means setting \( x^3 + 8 = 0 \).
• Solve the equation. Here, \( x^3 = -8 \), leading to \( x = -2 \).

Thus, the x-intercept is at the point \((x = -2, y = 0)\). It’s crucial to ensure that the denominator is not zero at this intercept so that we have a valid solution. This means we avoid division by zero, keeping the function defined.

Y-Intercepts

The y-intercept provides important insights into a rational function’s graph. It tells where the function crosses the y-axis, which occurs when the input, \( x \), equals zero. To find the y-intercept for the function \( r(x)=\frac{x^{3}+8}{x^{2}+4} \), use the following approach:

• Substitute \( x = 0 \) directly into the function.
• Calculate the resulting expression. For this function, substituting gives \( r(0) = \frac{0^3 + 8}{0^2 + 4} = \frac{8}{4} = 2 \).

This means the y-intercept is at \((x = 0, y = 2)\). The y-intercept is straightforward to find as you simply replace the input with zero and compute the result. This point represents where the function’s graph intersects the y-axis, providing a starting point for graphing or sketching out the rest of the function behavior.

Numerator and Denominator

The structure of a rational function, like \( r(x)=\frac{x^{3}+8}{x^{2}+4} \), is hinged on its numerator and denominator. Comprehending how these components affect the overall behavior of the function is vital:

• Numerator: The numerator, \( x^3 + 8 \), determines the x-intercepts. Setting it to zero gives the values of \( x \) where the function's output is zero, facilitating the finding of x-intercepts.
• Denominator: The denominator, \( x^2 + 4 \), plays a critical role in defining where the function is undefined. The function is not valid when this part equals zero, as it would imply division by zero.
• Double-checking at critical points like x-intercepts ensures the denominator is never zero ensuring those points are real and valid on the graph.

Together, the numerator and denominator dictate not only the intercepts but also highlight potential asymptotes and undefined regions, key for plotting and fully understanding rational functions.