Problem 55 - Algebra 2

Question

Determine any vertical asymptotes and holes in the graph of each rational function. $$ f(x)=\frac{x+2}{x^{2}+3 x-4} $$

Step-by-Step Solution

Answer

Vertical asymptotes at \( x = -4 \) and \( x = 1 \); no holes in the graph.

Step 1: Factor the Denominator

First, factor the quadratic expression in the denominator, which is \( x^2 + 3x - 4 \). By inspection or using the quadratic formula, we find the factors to be \( (x + 4)(x - 1) \). Thus, \( x^2 + 3x - 4 = (x + 4)(x - 1) \).

Step 2: Analyze for Holes and Vertical Asymptotes

The rational function is \( f(x) = \frac{x + 2}{(x + 4)(x - 1)} \). To find holes, check if any factor in the denominator also appears in the numerator and cancels out, but here \( x + 2 \) does not cancel with \( (x + 4)(x - 1) \), so there are no holes.

Step 3: Identify Vertical Asymptotes

Set the factors of the denominator equal to zero to find where the function is undefined. Thus, we solve \( x + 4 = 0 \) and \( x - 1 = 0 \), giving \( x = -4 \) and \( x = 1 \). These are the values of \( x \) that create vertical asymptotes.

Key Concepts

Vertical Asymptotes

A vertical asymptote is a line that a graph approaches but never actually touches or crosses. It occurs where the rational function is undefined due to division by zero. In the function \( f(x) = \frac{x + 2}{(x + 4)(x - 1)} \), vertical asymptotes can be found by setting the denominator equal to zero and solving for \( x \).

• For the factor \( (x + 4) \), solving \( x + 4 = 0 \) gives \( x = -4 \).
• For the factor \( (x - 1) \), solving \( x - 1 = 0 \) gives \( x = 1 \).

These values of \( x \) indicate the location of vertical asymptotes: they are \( x = -4 \) and \( x = 1 \). Vertical asymptotes tell us about the function's behavior near those \( x \)-values, where it shoots off towards infinity or negative infinity.

Holes in Graphs

Holes in graphs occur when there is a common factor in both the numerator and the denominator of a rational function. This common factor leads to cancellation, creating a "hole," or a point where the function is not defined even though the graph appears to pass through it. For the function \( f(x) = \frac{x + 2}{(x + 4)(x - 1)} \), we look for factors that cancel between the numerator and the denominator.In our given function, \( x + 2 \) does not factor or cancel out with \( (x + 4) \) or \( (x - 1) \). Therefore, there are no common factors to cancel out, which means there are no holes in the graph of this function. When you have a hole, it's like a missing dot on the graph where the function should have had a value but doesn't.

Factoring Polynomials

Factoring polynomials is a crucial skill in understanding and simplifying rational functions. If the polynomial in the denominator can be factored, it becomes much easier to identify any vertical asymptotes and hole locations in the graph.In the function \( f(x) = \frac{x + 2}{x^2 + 3x - 4} \), the first step in analyzing its graph is factoring the quadratic expression in the denominator. The polynomial \( x^2 + 3x - 4 \) can be factored into \( (x + 4)(x - 1) \).Factoring involves finding two binomials that multiply together to give you the original quadratic. Here's a simple step-by-step approach:

• Examine the polynomial for easy factors or use methods like grouping or the quadratic formula if it's not easily factorable by inspection.
• Check your factorization by multiplying back to ensure it equals the original polynomial.

In this case, factoring reveals the roots or zeros needed to find vertical asymptotes and possible holes, simplifying the analysis of the rational function.