Problem 55 - Precalculus: Functions and Graphs

Question

\( Graph f,\) and find equations of the vertical asymptotes. $$f(x)=\frac{20 x^{2}+80 x+72}{10 x^{2}+40 x+41}$$

Step-by-Step Solution

Answer

No vertical asymptotes exist for \\(f(x)\\) as the denominator has no real roots.

Step 1: Understand Vertical Asymptotes Definition

Vertical asymptotes occur where the function \(f(x)\) is undefined due to division by zero in the denominator. For the function \(f(x) = \frac{20x^2 + 80x + 72}{10x^2 + 40x + 41}\), we need to find the values of \(x\) that make the denominator zero.

Step 2: Set the Denominator Equal to Zero

Let's solve the equation \(10x^2 + 40x + 41 = 0\) to find potential vertical asymptotes. We need to check if there are real roots by using the quadratic formula to solve for \(x\).

Step 3: Apply the Quadratic Formula

To solve \(10x^2 + 40x + 41 = 0\), use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 10\), \(b = 40\), and \(c = 41\). Compute the discriminant \(b^2 - 4ac\) first.

Step 4: Calculate the Discriminant

Compute \(b^2 - 4ac = 40^2 - 4 \cdot 10 \cdot 41\). Calculate this to find \(1600 - 1640 = -40\). Since the discriminant is negative, the quadratic has no real solutions.

Step 5: Conclude About Vertical Asymptotes

Since the discriminant of the denominator \(10x^2 + 40x + 41\) yields no real solutions, there are no real values of \(x\) that make the denominator zero. Therefore, there are no vertical asymptotes for \(f(x)\).

Key Concepts

Quadratic Formula

The quadratic formula is a reliable method used for finding solutions to a quadratic equation of the form \[ax^2 + bx + c = 0\]If you're questioning why it always works, here is a simple breakdown. It helps in finding the roots by using the following formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula is universally applicable for any quadratic equation. Here's how you use it:

• Identify the coefficients, \(a\), \(b\), and \(c\) from your equation.
• Substitute these into the formula.
• Solve for \(x\).

The quadratic formula can give you all potential roots a quadratic equation might have, whether real or complex.

Discriminant

The discriminant is a part of the quadratic formula that can instantly tell you the nature of the roots without needing a full solution. It is the expression under the square root in the quadratic formula:\[b^2 - 4ac\]
Here's how it works:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root.
• If it is negative, the equation has no real roots but has two complex roots instead.

The discriminant is useful because it shows whether real solutions exist or not, saving computational effort in cases like our example where it turns out negative.

Rational Functions

Rational functions are a type of function represented as the quotient of two polynomials. A general form of a rational function is:
\[f(x) = \frac{P(x)}{Q(x)}\]
where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not equal to zero. Here are some characteristics of rational functions:

• Vertical asymptotes can occur where the denominator equals zero and the numerator is not zero.
• Horizontal asymptotes can describe the behavior of the function as \(x\) approaches infinity.

In our function, the numerator and denominator are polynomials of degree 2, making it a rational function.

Undefined Function

An undefined function arises in cases where the denominator equals zero. As division by zero is undefined in mathematics, this means that anywhere the denominator of a rational function is zero, the function is also undefined.
Vertical asymptotes are indicators of such undefined points within the context of the rational functions. However, if solving for when the denominator is zero results in complex numbers, as indicated by a negative discriminant in our exercise, no real 'undefined points' or vertical asymptotes exist. This means for\[f(x) = \frac{20x^2 + 80x + 72}{10x^2 + 40x + 41}\]
there will be no real values of \(x\) that make \(f(x)\) undefined because the roots for the denominator are not real.