Problem 11 - Precalculus Mathematics for Calculus

Question

Find the \(x\) - and \(y\) -intercepts of the rational function. $$r(x)=\frac{x-1}{x+4}$$

Step-by-Step Solution

Answer

The \(x\)-intercept is \((1, 0)\) and the \(y\)-intercept is \((0, -\frac{1}{4})\).

Step 1: Identify A Rational Function

The function given is a rational function, which is a fraction with polynomials in the numerator and the denominator: \(r(x) = \frac{x-1}{x+4}\). Our task is to find both the \(x\)-intercepts and the \(y\)-intercepts of this function.

Step 1: Solve for the x-intercept

To find the \(x\)-intercept, set the numerator equal to zero and solve for \(x\). This is because the function equals zero when the numerator is zero.\[x - 1 = 0\]Solve this equation:\[x = 1\]Therefore, the \(x\)-intercept is at \((1, 0)\).

Step 2: Solve for the y-intercept

To find the \(y\)-intercept, substitute \(x = 0\) into the function \(r(x)\) and solve for \(y\).\[r(0) = \frac{0-1}{0+4} = \frac{-1}{4}\]Thus, the \(y\)-intercept is at \(\left(0, -\frac{1}{4}\right)\).

Key Concepts

x-intercepts

The x-intercept of a rational function is the point where the graph of the function crosses the x-axis. At this point, the value of the function, or the 'output,' is zero, meaning we write it as

• the ordered pair \((x, 0)\).

To find the x-intercept of a rational function like \(r(x) = \frac{x-1}{x+4}\),we focus on the numerator of the function.

• Set the numerator equal to zero: \(x - 1 = 0\).
• Solve this equation to find the value of \(x\):\(x = 1\).

Thus, the x-intercept of the function \(r(x)\) is at the point \((1, 0)\).Notice how only the numerator affects the x-intercept. This is because setting the function to zero neutralizes the effect of the denominator.

y-intercepts

The y-intercept is where the graph of the function intersects the y-axis. Unlike the x-intercept, at this point, the input \(x\) is zero and the output is the y-value. Hence, we express this point as

• the ordered pair \((0, y)\).

To determine the y-intercept of a rational function like \(r(x) = \frac{x-1}{x+4}\),substitute \(x = 0\) into the function:

• With \(x = 0\), calculate \(r(0) = \frac{0-1}{0+4} = \frac{-1}{4}\).

Therefore, the y-intercept is at \((0, -\frac{1}{4})\).Finding the y-intercept gives us a clear picture of where the graph starts on the y-axis. It's crucial for understanding the starting point of the graph.

numerator and denominator

In a rational function, the numerator and denominator are key components. Think of a rational function as a fraction \(\frac{a(x)}{b(x)}\), where both \(a(x)\) and \(b(x)\)are polynomials.
The numerator, which is the top part of the fraction, dictates where the graph crosses the x-axis. It is crucial in finding the x-intercepts because when \(a(x) = 0\), the whole fraction equals zero.
For the function \(r(x) = \frac{x-1}{x+4}\),

• the numerator is \(x - 1\), and
• it determines the x-intercept as discussed in the previous section.

On the other hand, the denominator \(b(x)\)influences restrictions, or values that \(x\) should not take.
For instance, the denominator should not be equal to zero because it would make the function undefined.
In \(r(x)\), the denominator \(x + 4\)indicates that \(x = -4\) would make the function undefined, thus affecting the domain. Understanding the numerator and denominator helps you better capture the behavior and limitations of the function.

polynomials

Polynomials are algebraic expressions made up of terms connected by addition or subtraction, consisting of variables raised to non-negative integer powers.
They appear in both the numerator and the denominator of a rational function. Knowing this is key when analyzing and graphing rational functions.
In the rational function \(r(x) = \frac{x-1}{x+4}\),

• both \(x - 1\) and \(x + 4\) are first-degree polynomials.
• First-degree polynomials, or linear polynomials, have graphs that are straight lines.

Looking at rational functions often involves inspecting these polynomial components separately to understand their intersections with the axes and how they influence the behavior of the graph.
One crucial thing to remember is that polynomial degrees can suggest the number of intercepts and the fundamental shape of their graphs. Linear polynomials, like those in our example, affect the graph in a simple, understandable manner, making the calculation of intercepts straightforward.