Problem 477 - Precalculus

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$f(x)=\frac{x^{2}+x+6}{x^{2}-10 x+24}$$

Step-by-Step Solution

Answer

No x-intercepts; y-intercept (0, 1/4).

Step 1: Find the x-intercepts

To find the x-intercepts, set the function equal to zero and solve for \(x\): \[f(x) = \frac{x^2 + x + 6}{x^2 - 10x + 24} = 0\]The numerator must equal zero: \[x^2 + x + 6 = 0\] Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 1\), \(b = 1\), and \(c = 6\), we calculate the discriminant: \[b^2 - 4ac = 1^2 - 4\cdot1\cdot6 = 1 - 24 = -23\]Since the discriminant is negative, the quadratic has no real roots. Thus, there are no x-intercepts.

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the function:\[f(0) = \frac{0^2 + 0 + 6}{0^2 - 10\cdot0 + 24} = \frac{6}{24} = \frac{1}{4}\]Thus, the y-intercept is \((0, \frac{1}{4})\).

Step 3: Summary

The function \(f(x) = \frac{x^2 + x + 6}{x^2 - 10x + 24}\) does not have any x-intercepts because the numerator has no real roots. The y-intercept is found at \((0, \frac{1}{4})\).

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool for finding the roots of a quadratic equation. A quadratic equation is any equation that can be transformed into the form \(ax^2 + bx + c = 0\). This formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula, you can find the values of \(x\) that make the quadratic equation zero, which are called the roots.

• \(a\), \(b\), and \(c\) are the coefficients of the equation.
• The \(\pm\) in the formula indicates that there are generally two possible solutions for \(x\).

In our exercise, the numerator was a quadratic equation \(x^2 + x + 6 = 0\), where \(a = 1\), \(b = 1\), and \(c = 6\). So, we used the quadratic formula to find its roots.

Discriminant

The discriminant is part of the quadratic formula, specifically the expression under the square root: \(b^2 - 4ac\). It tells us a lot about the nature of the roots of the quadratic equation. Here's what the discriminant indicates:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, also known as a double root.
• If \(b^2 - 4ac < 0\), there are no real roots, only complex ones.

In the exercise, the discriminant was calculated as \(-23\), which is negative. This means that the quadratic has no real roots, indicating why the function \(f(x)\) had no x-intercepts.

Real Roots

Real roots are the solutions of a quadratic equation that are real numbers. These roots can be found using the quadratic formula, as long as the discriminant (\(b^2 - 4ac\)) is non-negative. Real roots are

• Where the graph of the quadratic equation touches or crosses the x-axis.
• Essential for determining the x-intercepts of a function.

In problems involving the sum of rational functions, finding the real roots of the numerator is key to locating the x-intercepts. However, in our specific exercise, since the discriminant was negative, there were no real roots and consequently no x-intercepts.

Rational Function

A rational function is defined as the ratio of two polynomials, where the numerator and the denominator are polynomial expressions. The function given in our exercise was \[f(x) = \frac{x^2 + x + 6}{x^2 - 10x + 24}\]To find the intercepts of a rational function:

• X-intercepts: Occur where the numerator itself equals zero (providing the denominator is not zero).
• Y-intercept: Found by evaluating the function at \(x = 0\).

For our exercise, the numerator did not produce real roots, leading to no x-intercepts. The y-intercept was calculated by plugging \(x = 0\) into the function and simplifying.