Problem 10 - 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

X-intercept: (-2, 0); Y-intercept: (0, 2).

Step 1: Find the x-intercepts

To find the x-intercepts of the function, we set the numerator equal to zero and solve for \(x\). The x-intercepts occur where the function crosses the x-axis, i.e., when \(r(x) = 0\). Thus, we solve: \[ x^3 + 8 = 0 \] \[ x^3 = -8 \] \[ x = oot[3]{-8} \] \[ x = -2 \] So, the x-intercept is \(x = -2\).

Step 2: Find the y-intercepts

To find the y-intercepts, substitute \(x = 0\) in the function \(r(x)\): \[ r(0) = rac{0^3 + 8}{0^2 + 4} = rac{8}{4} = 2 \] So, the y-intercept is \(r(x) = 2\).

Step 3: Summarize the Intercepts

Based on the solutions from the previous steps, the x-intercept is at \((-2,0)\), and the y-intercept is at \((0,2)\).

Key Concepts

Understanding x-intercepts in Rational Functions

When dealing with rational functions, understanding how to find the x-intercepts is essential. The x-intercept of a function is where the graph crosses the x-axis. At these points, the value of the function is zero. For rational functions, like our example function \( r(x) = \frac{x^3 + 8}{x^2 + 4} \), we need to focus on the numerator.

To find the x-intercepts:

• Set the numerator equal to zero: \( x^3 + 8 = 0 \).
• Solve for \( x \) by isolating the variable: \( x^3 = -8 \).
• Take the cube root to find \( x \): \( x = \sqrt[3]{-8} \).
• Conclude with the x-intercept: \( x = -2 \).

This point \((-2, 0)\) is crucial as it represents where the graph touches the x-axis. Always remember that for rational functions, x-intercepts come from setting the numerator to zero unless the value is excluded by the denominator.

Determining y-intercepts of Rational Functions

The y-intercept is where the function crosses the y-axis, meaning the x-value is zero. For a rational function, finding this point involves a straightforward substitution.

To find the y-intercepts in the given function \( r(x) = \frac{x^3 + 8}{x^2 + 4} \):

• Substitute \( x = 0 \) into the function: \( r(0) = \frac{0^3 + 8}{0^2 + 4} \).
• Simplify, calculating the result: \( r(0) = \frac{8}{4} = 2 \).
• The result gives us the y-intercept: \( r(x) = 2 \).

So, the y-intercept of the function is \((0, 2)\). This information is essential, as it indicates how the function interacts with the y-axis. Understanding the y-intercept helps provide a clearer picture of the overall graph of the function.

Comprehensive Function Analysis

Analyzing rational functions involves considering their intercepts as well as their general behavior. By identifying both x- and y-intercepts, you can begin to visualize the shape of the graph.

For the given function \( r(x) = \frac{x^3 + 8}{x^2 + 4} \):

• You have identified the x-intercept at \((-2, 0)\).
• The y-intercept is at \((0, 2)\).

Besides intercepts, other aspects may include asymptotes and the domain, but they aren't required for this problem. Understanding the intercepts offers a starting point for plotting the function or estimating its behavior. Graphically, intercepts serve as reference points and aid in sketching. Furthermore, always consider these points in the broader context of how a rational function behaves compared to linear or polynomial graphs. This makes intercepts powerful tools in predicting and understanding the trajectory of the function's graph.