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Chapter 4: Problem 12

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

Short Answer

Expert verified
Both the x-intercept and y-intercept are at (0,0).

Step by step solution

01

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 \((0,0)\).
02

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 \((0,0)\).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(\mathrm{a}) .\) $$ P(x)=2 x^{4}+15 x^{3}+17 x^{2}+3 x-1 $$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=6 x^{4}-7 x^{3}-8 x^{2}+5 x $$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=4 x^{4}-21 x^{2}+5 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{x^{3}+x^{2}}{x^{2}-4} $$

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=3 x^{3}+17 x^{2}+21 x-9 $$
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