Problem 6 - Algebra and Trigonometry

Question

Find the \(x\) -and \(y\) -intercepts of the rational function. \(s(x)=\frac{3 x}{x-5}\)

Step-by-Step Solution

Answer

Both intercepts are at (0,0).

Step 1: Find the x-intercept

To find the x-intercept of the function \(s(x) = \frac{3x}{x-5}\), set \(s(x)\) to zero and solve for \(x\). This means \(\frac{3x}{x-5} = 0\). Since a fraction is zero when its numerator is zero, set the numerator to zero: \(3x = 0\). Solving for \(x\) gives \(x = 0\). Therefore, the x-intercept is \((0,0)\).

Step 2: Find the y-intercept

To find the y-intercept of \(s(x)\), substitute \(x = 0\) into the function. This yields \(s(0) = \frac{3(0)}{0-5} = 0\). Thus, the y-intercept is also \((0,0)\).

Key Concepts

x-intercepts

The x-intercepts of a rational function are the points where the graph of the function crosses the x-axis. These intercepts occur where the function's value is zero. For a rational function, which takes the form \( f(x) = \frac{p(x)}{q(x)} \), the x-intercepts are found by setting the numerator \( p(x) \) equal to zero and solving for \( x \).

In our problem, the function is \( s(x) = \frac{3x}{x-5} \). To find the x-intercepts, we set the numerator, \( 3x \), equal to zero:

- Solve for \( x \): - \( 3x = 0 \) - Dividing both sides by 3 gives \( x = 0 \).

So, the x-intercept of the function is at \( (0,0) \). Remember that only the numerator determines the x-intercepts because a fraction is zero only when its numerator is zero.

y-intercepts

Y-intercepts are the points where the graph of a function crosses the y-axis. These occur when the input variable \( x \) is zero, and we are interested in the value of the function at this point. To find a y-intercept for any function \( f(x) \), substitute \( x = 0 \) into the equation and solve for \( f(x) \).

For the function \( s(x) = \frac{3x}{x-5} \), we substitute zero in for \( x \):

• Replace \( x \) with 0: \( s(0) = \frac{3(0)}{0-5} \)
• This simplifies to \( s(0) = \frac{0}{-5} = 0 \)

This calculation shows that the y-intercept is \( (0,0) \), the same coordinate as the x-intercept. In rational functions, sometimes x and y intercepts coincide at the origin, but this is not always the case.

numerator and denominator of a fraction

Understanding the role of the numerator and denominator in a rational function is crucial for grasping how these functions behave. A rational function is expressed as a fraction, \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials.

Here's the core idea:

• The numerator \( p(x) \) determines where the rational function is equal to zero, thus helping us find x-intercepts.
• The denominator \( q(x) \) determines where the function is undefined, which occurs where \( q(x) = 0 \), possibly creating vertical asymptotes, but these do not affect where the function crosses axes.

In the given function \( s(x) = \frac{3x}{x-5} \),
- The numerator \( 3x \) becomes zero at \( x = 0 \), leading to the x-intercept at \( (0,0) \).
- The denominator \( x-5 \) is zero at \( x = 5 \), where the function is undefined, indicating a vertical asymptote.

When solving for intercepts, focus on the numerator to understand when the entire fraction equals zero. This principle simplifies finding intercepts in rational functions.