Problem 53 - Algebra and Trigonometry

Question

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. \(r(x)=\frac{3 x^{2}+6}{x^{2}-2 x-3}\)

Step-by-Step Solution

Answer

No x-intercept, y-intercept is -2, vertical asymptotes at x = 3 & x = -1, horizontal asymptote at y = 3.

Step 1: Find the X-Intercepts

To find the x-intercepts of the function, we set the numerator equal to zero because an x-intercept occurs where the function value is zero:\[ 3x^2 + 6 = 0 \]Solving this equation:\[ 3x^2 = -6 \]\[ x^2 = -2 \]Since there is no real number solution where a square equals a negative, the function has no x-intercepts.

Step 2: Find the Y-Intercept

To find the y-intercept, we set \(x = 0\) and solve for \(r(x)\):\[ r(0) = \frac{3(0)^2 + 6}{(0)^2 - 2(0) - 3} = \frac{6}{-3} = -2 \]Therefore, the y-intercept of the function is \(y = -2\).

Step 3: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator does not equal zero simultaneously. Set the denominator equal to zero and solve:\[ x^2 - 2x - 3 = 0 \]Factoring the quadratic:\[ (x - 3)(x + 1) = 0 \]Thus, the vertical asymptotes are at \(x = 3\) and \(x = -1\).

Step 4: Find the Horizontal Asymptote

For rational functions, the horizontal asymptote is determined by the degrees of the numerator and the denominator. Since both have degree 2:- The horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are coefficients of the highest degree terms, which are 3 and 1 respectively.So, the horizontal asymptote is \(y = \frac{3}{1} = 3\).

Step 5: Sketch the Graph

Using the calculated intercepts and asymptotes, draw the function:- Plot the y-intercept at \( (0, -2) \).- Draw vertical lines at \(x = 3\) and \(x = -1\) for asymptotes.- Draw a horizontal line at \(y = 3\) for the horizontal asymptote.The graph will approach these asymptotes but never touch them. Use these guidelines to sketch a rough graph of \(r(x)\).

Key Concepts

X-Intercepts

Let's dive into finding the x-intercepts of a rational function. This occurs where the graph of the function crosses the x-axis, meaning the output of the function, or the y-value, is zero at this point. To find any x-intercepts, we set the numerator of the rational function equal to zero and solve for x. This is because a fraction is zero only when its numerator is zero.

For the function in our exercise, \( r(x) = \frac{3x^2 + 6}{x^2 - 2x - 3} \), we set the numerator \( 3x^2 + 6 = 0 \). Solving \( 3x^2 = -6 \) results in \( x^2 = -2 \). But, there's no real number that when squared gives a negative result, meaning the function has no x-intercepts.

This tells us an important detail about the graph: it doesn't touch or cross the x-axis. Hence, in this case, no x-intercepts exist.

Y-Intercepts

Next up is finding the y-intercept, which is where the function crosses the y-axis. To find the y-intercept of any rational function, simply substitute zero for x in the function and solve for y. This tells you the point where the graph crosses the y-axis, which can be incredibly helpful for sketching the graph.

For our function, substitute \(x = 0\):

• Compute: \( r(0) = \frac{3(0)^2 + 6}{(0)^2 - 2(0) - 3} \)
• Simplified: \( r(0) = \frac{6}{-3} = -2 \)

Therefore, the y-intercept is at \(y = -2\). This tells us the rational function graph will intersect the y-axis at the point (0, -2).

Remembering intercepts helps us quickly identify key graphing points and understand the overall behavior of the function.

Vertical Asymptotes

Vertical asymptotes occur where the function is undefined, which happens when the denominator of the rational function equals zero, and the numerator is non-zero at the same time. These are the lines that the graph can approach infinitely close but never touch or cross.

For \( r(x) = \frac{3x^2 + 6}{x^2 - 2x - 3} \), we set the denominator \( x^2 - 2x - 3 = 0 \) and solve:

• Factor: \((x - 3)(x + 1) = 0\)
• Solve: \(x = 3\) or \(x = -1\)

Thus, vertical asymptotes are present at \(x = 3\) and \(x = -1\). This means the graph will have these 'invisible barriers' that it will never cross, extending indefinitely along these vertical lines.

These help define the function's shape and behavior across different x-values and are crucial for understanding its logic.

Horizontal Asymptotes

Horizontal asymptotes occur in rational functions and describe the end behavior of a graph, or how it behaves as x goes to infinity or negative infinity. To find a horizontal asymptote, observe the degree of the polynomial in both the numerator and the denominator.

When both polynomials have the same degree, the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator. In our function:

• Top degree: \(3x^2\)
• Bottom degree: \(x^2\)
• Coefficients: \(3\) from the numerator, \(1\) from the denominator

Thus, the horizontal asymptote is at \(y = \frac{3}{1} = 3\). This implies that as x moves towards infinity or negative infinity, the value of \(r(x)\) stabilizes around 3.

Understanding horizontal asymptotes helps in visualizing how the graph stretches towards extreme x-values and gives key insights into the function’s long-term trends.