Problem 69 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function involving common factors and find all intercepts and asymptotes. Indicate all asymptotes on the graph. $$f(x)=\frac{x^{2}+3 x-10}{x-2}$$

Step-by-Step Solution

Answer

The graph is a straight line y = x + 5 with a hole at x = 2. The x and y intercepts are at x = - 5 and y = 5, respectively, and there is a vertical asymptote at x = 2.

Step 1: Factor the numerator

Factoring the numerator: \(x^{2}+3 x-10\) can be factored into \((x - 2)(x + 5)\)

Step 2: Simplify the function

Simplify the function by cancelling the common factors (x - 2) in the numerator and denominator. This gives: f(x) = x + 5, except x ≠ 2

Step 3: Determine and plot intercepts

For x-intercept, set f(x) = 0 -> x = - 5. For y-intercept, set x=0, g(x) = 5. Now plot the intercepts on the graph.

Step 4: Determine and plot vertical asymptote

The original function is undefined at x = 2, thus x = 2 is a vertical asymptote. Plot this asymptote on the graph.

Step 5: Draw the graph

Sketch the line using the y-intercept and the x-intercept, and make a hole at x = 2 because of the asymptote. The graph appears as the line y = x + 5, except that there is a hole at x = 2.

Key Concepts

Understanding Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches. In rational functions, asymptotes often occur where the function is undefined.
In our exercise, we have a vertical asymptote at \(x = 2\), which is where the original function \(f(x)=\frac{x^{2}+3x-10}{x-2}\) becomes undefined. This happens because we end up dividing by zero when \(x = 2\).
Whenever you're working with rational functions, finding vertical asymptotes is crucial. You do this by setting the denominator equal to zero and solving for \(x\).
The importance of asymptotes is in showing where the function's behavior changes dramatically. At \(x = 2\), the graph of our modified function \(f(x) = x + 5\) has a hole instead of a continuation. This hole represents the asymptote's original position, showcasing the disruption in continuity.

The Role of Factorization

Factorization is the process of breaking down a complex expression into simpler products. This is especially useful in rational functions as it helps in simplifying expressions.
In the given problem, the numerator \(x^{2}+3x-10\) is factored into \((x - 2)(x + 5)\). This factorization reveals a common factor with the denominator \(x - 2\).
Once you factor both the numerator and the denominator, you can simplify the expression by canceling out common factors. This step is essential for simplifying rational expressions, as it helps you easily identify intercepts and asymptotes.
Remember to always state the domain of the function carefully especially after canceling factors as this might lead to points where the function remains undefined, like at \(x = 2\) in our problem.

Finding Intercepts

Intercepts are points where the graph crosses the axes. These points can give significant insight into the behavior of a function.
For x-intercepts, solve the equation \(f(x) = 0\). This is done by finding the value of \(x\) that makes the numerator zero, while making sure this \(x\) value doesn't make the denominator zero as well.

• In our function, set \(x+5 = 0\), solving this we get \(x = -5\), which is our x-intercept.

For y-intercepts, simply plug in \(x = 0\) into your simplified function. The output will be the y-intercept, representing where the graph crosses the y-axis.

• In this exercise, substituting \(x=0\) into \(f(x) = x + 5\) results in \(f(x) = 5\). This gives us a y-intercept at \(y = 5\).

Identifying intercepts helps in sketching the graph accurately and understanding the function's properties.