Problem 12 - 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 the x-intercept and y-intercept are at (0,0).

Step 1: Identifying the x-intercept

To find the x-intercept of the function \( s(x) = \frac{3x}{x-5} \), set the numerator equal to zero. The x-intercept occurs where the function equals zero.Set \( 3x = 0 \). Solve for \( x \).\( x = 0 \). So, the x-intercept is at \((0,0)\).

Step 2: Identifying the y-intercept

To find the y-intercept, substitute \( x = 0 \) into the function \( s(x) = \frac{3x}{x-5} \). This will give the value of the function at \( x = 0 \).\[ s(0) = \frac{3 \times 0}{0 - 5} = \frac{0}{-5} = 0 \]So, the y-intercept is at \((0,0)\).

Step 3: Confirming Intercepts

Both the x-intercept and the y-intercept are located at the origin \((0,0)\). This suggests that the graph of the function \( s(x) = \frac{3x}{x-5} \) passes through the origin consistent with our calculations.

Key Concepts

Understanding X-intercept

To find the x-intercept of a rational function like \( s(x) = \frac{3x}{x-5} \), look where the function crosses the x-axis. This happens when the output \( s(x) \) is zero.
This means we need to set the numerator equal to zero since the whole fraction becomes zero when the top part (numerator) is zero.

• The numerator of \( s(x) = \frac{3x}{x-5} \) is \( 3x \).
• Set \( 3x = 0 \).
• Solve for \( x \) to get \( x = 0 \).

Therefore, the x-intercept is at the point \((0,0)\), indicating that the graph passes through the origin on the x-axis.
It’s important to understand that finding the x-intercept helps us know where the function touches the x-axis.

Understanding Y-intercept

To find the y-intercept, determine where the graph crosses the y-axis. This occurs when \( x \) is zero in the function.
Simply plug \( x = 0 \) into \( s(x) = \frac{3x}{x-5} \) and solve for \( s(0) \). Here's a step by step way to do it:

• Substitute \( x = 0 \) into the function \( s(x) = \frac{3x}{x-5} \).
• This gives \( s(0) = \frac{3 \times 0}{0 - 5} = \frac{0}{-5} = 0 \).

The calculation shows that the y-intercept is \((0,0)\) as well.
This means the graph intersects the y-axis at the origin, which reveals a special point where both x- and y- intercepts are the same.

Overview of Function Graph Analysis

Analyzing the graph of a rational function like \( s(x) = \frac{3x}{x-5} \) involves understanding different properties, including its intercepts and asymptotes.
For the function, the intercepts discovered tell us significant points for graph plotting:

• The x-intercept is \((0,0)\).
• The y-intercept is also \((0,0)\).

Since the function contains a denominator \( x-5 \), this shows a vertical asymptote at \( x = 5 \). It means as \( x \) approaches 5, the function becomes infinitely large or small, and the graph will never touch this line.
The intercepts at the origin and vertical asymptote together shape how the graph behaves, giving us a comprehensive view of the function's graph.