Problem 25 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) $$f(x)=\frac{3 x^{2}-3 x-36}{x^{2}+x-2}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = -2\) and \(x = 1\); horizontal asymptote at \(y = 3\); x-intercepts at \(x = 4\) and \(x = -3\).

Step 1: Factor the Numerator and Denominator

First, we need to factor both the numerator and the denominator. The numerator is \(3x^2 - 3x - 36\). We can factor out a 3, so it becomes \(3(x^2 - x - 12)\). The trinomial \(x^2 - x - 12\) factors into \((x-4)(x+3)\). Thus, the numerator is \(3(x-4)(x+3)\). For the denominator \(x^2 + x - 2\), it factors into \((x+2)(x-1)\). Thus, the function is \(f(x) = \frac{3(x-4)(x+3)}{(x+2)(x-1)}\).

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur when the denominator equals zero. We solve \((x+2)(x-1)=0\), which gives \(x = -2\) and \(x = 1\). These are the vertical asymptotes of the function.

Step 3: Identify the Horizontal Asymptote

For horizontal asymptotes, compare the degrees of the numerator and denominator. Both are degree 2. The asymptote is the ratio of leading coefficients, which is \(\frac{3}{1} = 3\). Thus, the horizontal asymptote is \(y = 3\).

Step 4: Find the X-Intercepts

X-intercepts occur when the numerator is zero. Solve \(3(x-4)(x+3)=0\), which gives \(x = 4\) and \(x = -3\). These are the x-intercepts of the function.

Step 5: Analyze Behavior Near Asymptotes

Check the behavior of \(f(x)\) as \(x\) approaches the vertical asymptotes \(x = -2\) and \(x = 1\). The function will approach positive or negative infinity at these points depending on the sign of \(f(x)\) near the asymptotes.

Step 6: Plot Key Points and Sketch the Graph

With all critical information, plot the x-intercepts \((-3, 0)\) and \((4, 0)\), vertical asymptotes at \(x = -2\) and \(x = 1\), and the horizontal asymptote \(y = 3\). Use this to sketch the overall behavior of the graph, considering the sign of \(f(x)\) in each interval around the asymptotes and intercepts.

Key Concepts

Graph Sketching

Sketching the graph of a rational function involves piecing together key information about the function's behavior and characteristics. Begin with factoring the numerator and denominator to simplify the function if possible. Factorization makes identifying asymptotes and intercepts much easier. Next, pinpoint the vertical and horizontal asymptotes, as these will guide the direction in which the graph moves as it extends indefinitely. Identify the x-intercepts, which are points where the graph intersects the x-axis. Additionally, consider the behavior of the function around these asymptotes and intercepts. Finally, plot these critical points and asymptotes on the graph. This way you can sketch the curves that represent the function's actual path, keeping in mind the intervals in which the function approaches infinity.

Vertical Asymptotes

Vertical asymptotes are straight vertical lines that the graph approaches but never touches. They occur in rational functions where the denominator is zero, as this leads to undefined values in the function. For the function \(f(x) = \frac{3(x-4)(x+3)}{(x+2)(x-1)}\), set the denominator \((x+2)(x-1)\) equal to zero and solve for \(x\). This gives the vertical asymptotes at \(x = -2\) and \(x = 1\). As \(x\) approaches these values, \(f(x)\) will tend towards infinity, either positive or negative, indicating rapid changes in the function's behavior just before crossing these lines. Evaluating the sign of \(f(x)\) on either side of these points helps determine whether the graph approaches the asymptote from the positive or negative side.

Horizontal Asymptote

Horizontal asymptotes describe the behavior of a function as \(x\) approaches positive or negative infinity. For the ratio of polynomials in our function, compare their degrees. If the degrees are the same, the horizontal asymptote is found by taking the ratio of the leading coefficients. Here, both the numerator and denominator are quadratic (degree 2), so the horizontal asymptote is given by \(y = \frac{3}{1} = 3\). This means that as \(x\) grows very large in either direction, the value of \(f(x)\) gets closer to 3. It's important to remember that while the function can cross its horizontal asymptote, unlike vertical asymptotes, the function will tend to stabilize and level out along this line as \(x\) extends to infinity.

X-Intercepts

X-intercepts are specific points where the graph crosses the x-axis, and they occur when the numerator of the rational function is zero and the function itself is well-defined. To find these for the function \(f(x) = \frac{3(x-4)(x+3)}{(x+2)(x-1)}\), set the numerator \(3(x-4)(x+3)\) equal to zero. Solving this equation gives the x-values 4 and -3, which means the x-intercepts are at \((4, 0)\) and \((-3, 0)\). These are the points you will plot on the graph where it touches or crosses the x-axis. It's crucial to ensure that these points do not coincide with a vertical asymptote, as the function is undefined at those points.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler components, making complex expressions easier to work with. For our function \(f(x) = \frac{3x^2 - 3x - 36}{x^2 + x - 2}\), begin by factoring the numerator and the denominator. Start with the numerator \(3x^2 - 3x - 36\). Factor out the common factor 3 to get \(3(x^2 - x - 12)\). Then, factor the quadratic \(x^2 - x - 12\) into \((x-4)(x+3)\). Similarly, factor the denominator \(x^2 + x - 2\) into \((x+2)(x-1)\). Simplifying these expressions makes it easier to analyze the behavior of the function, such as finding asymptotes and intercepts. Factoring is a crucial skill in algebra that helps in simplifying, solving, and understanding rational functions.