Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$ f(x)=\frac{x^{2}+8 x+7}{x^{2}+11 x+30} $$

Step-by-Step Solution

Answer

The x-intercepts are (-1, 0) and (-7, 0); the y-intercept is (0, 7/30).

Step 1: Understand the Problem

We need to find the intercepts of the given rational function: \( f(x) = \frac{x^2 + 8x + 7}{x^2 + 11x + 30} \). An intercept is a point where the graph of the function crosses the axes.

Step 2: Find the y-intercept

To find the \( y \)-intercept, set \( x = 0 \) and solve for \( f(x) \). This is because the \( y \)-intercept is where the graph crosses the \( y \)-axis, which occurs when \( x = 0 \): \[ f(0) = \frac{0^2 + 8 \cdot 0 + 7}{0^2 + 11 \cdot 0 + 30} = \frac{7}{30} \]Therefore, the \( y \)-intercept is \( (0, \frac{7}{30}) \).

Step 3: Find the x-intercepts

To find the \( x \)-intercepts, set \( f(x) = 0 \). The function equals zero when the numerator is zero and the denominator is not zero. Set the numerator equal to zero:\[ x^2 + 8x + 7 = 0 \]Factor the quadratic:\[ (x + 1)(x + 7) = 0 \]Solve for \( x \):\[ x + 1 = 0 \quad \text{or} \quad x + 7 = 0 \]\[ x = -1 \quad \text{or} \quad x = -7 \]The \( x \)-intercepts are \( (-1, 0) \) and \( (-7, 0) \).

Step 4: Validate the Denominator

For the \( x \)-intercepts found, ensure these values do not make the denominator zero. The denominator is zero when:\[ x^2 + 11x + 30 = 0 \]Factor the quadratic:\[ (x + 5)(x + 6) = 0 \]Solutions are \( x = -5 \) and \( x = -6 \). Since \(-1\) and \(-7\) do not equal \(-5\) or \(-6\), the found \( x \)-intercepts are valid.

Key Concepts

X-Intercepts

The concept of x-intercepts pertains to points where the graph of a function crosses the x-axis. To find these, you need to determine the x-values for which the function is equal to zero. In other words, solve the equation when the function's output (\( f(x)\) ) is zero.

In the case of rational functions, like\(f(x) = \frac{x^2 + 8x + 7}{x^2 + 11x + 30}\), x-intercepts occur when the numerator equals zero while the denominator is non-zero. This avoids undefined points.

Following the given solution:

• The numerator:\(x^2 + 8x + 7 = 0\) has to be solved. Factor this into:\((x + 1)(x + 7) = 0\)
• Solve each factor set to zero, giving solutions:\(x = -1\) and\(x = -7\)Thus, x-intercepts are at\((-1, 0)\) and\((-7, 0)\)

Ensure these values don't zero out the denominator. Confirm that these x-values do not make:\(x^2 + 11x + 30 = 0\), which factored is\((x + 5)(x + 6) = 0\). The valid solutions ensure safe calculation and correct result.

Y-Intercepts

Understanding y-intercepts is key. They represent where the graph crosses the y-axis, observed by setting\(x = 0\) in the function. This is because the y-axis is essentially the line where all x-values are zero.

Example:\(f(x) = \frac{x^2 + 8x + 7}{x^2 + 11x + 30}\) allows finding the y-intercept by substituting x with 0 in the function.

The calculation becomes:\[f(0) = \frac{0^2 + 8 \times 0 + 7}{0^2 + 11 \times 0 + 30} = \frac{7}{30}.\]Thus, the y-intercept is the point\( (0, \frac{7}{30})\).

• It provides a clear point on the graph to begin sketching.
• Only one y-intercept exists if the function is not piecewise.

Recognizing these intercepts aids in understanding the function's behavior and graph.

Quadratic Equations

Quadratic equations arise when working with polynomials and are pivotal in finding intercepts. In standard form, a quadratic equation is:\(ax^2 + bx + c = 0\). Such an equation can often be solved by factoring, completing the square, or applying the quadratic formula.

In our rational function example, the numerator\(x^2 + 8x + 7\) and the denominator\(x^2 + 11x + 30\) are both quadratic equations.

For factoring,

• Identify two numbers that multiply to the constant term and add to the linear coefficient.
• For\(x^2 + 8x + 7\), the factors are:\((x + 1)(x + 7)\).
• This simple technique enables simplifying or solving quadratics effectively.

Understanding these methods ensures adeptness in tackling rational functions.

Rational Functions

Rational functions are expressions represented as the quotient of two polynomials, making them crucial in higher-level algebra. Formally, a rational function is:\(f(x) = \frac{P(x)}{Q(x)}\), where\(P(x)\) and\(Q(x)\) are polynomials.

Key aspects include:

• Points where\(Q(x) = 0\) are crucial as they can indicate undefined points or vertical asymptotes in a graph.
• Finding intercepts involves examining both numerators and denominators of the rational function.
• Behavior near intercepts is essential for sketching graphs accurately.

Our function,\(f(x) = \frac{x^2 + 8x + 7}{x^2 + 11x + 30}\), serves as an example, highlighting how to identify x- and y-intercepts in this context.

Gain proficiency by practicing discerning such points and understanding their implications on graphs.