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When studying rational functions, finding intercepts is a fundamental skill.

Intercepts can be found as both:

• **x-intercepts**, where the graph of the function crosses the x-axis, and
• **y-intercepts**, where it crosses the y-axis.

Finding these points helps in graphing and understanding the behavior of the function.

x-Intercepts

To find the x-intercepts of a rational function, set the numerator equal to zero, as these are the values of x where the function's output, or y-value, is zero. However, in the case of the function \( r(x) = \frac{2}{x^2 + 3x - 4} \), the numerator is 2, a constant. Since 2 is never zero, this function thus has no x-intercepts.

y-Intercepts

For the y-intercepts, substitute \( x = 0 \) into the function. Simplify to find \( y \). Upon doing this with \( r(x) \), you'll find \( r(0) = -\frac{1}{2} \). Therefore, the y-intercept is located at \( (0, -\frac{1}{2}) \). This point tells us how the function behaves when it starts or ends across vertical lines in a graph.

To make sense of many rational functions, factoring is often an essential first step.

Quadratic expressions like \( x^2 + 3x - 4 \) need factoring to identify critical points and simplify the problem overall. Let's break it down:

Identify Possible Factors

Start by examining the equation \( x^2 + 3x - 4 \).

• Look for two numbers that multiply to \( -4 \) (the constant term) and add to \( 3 \) (the linear coefficient).
• These numbers are \( 4 \) and \( -1 \), because \( 4 \times -1 = -4 \) and \( 4 + (-1) = 3 \).

Form the Factored Expression

Using these numbers, rewrite the quadratic as \( (x + 4)(x - 1) \).
Factoring helps us understand the function better, making it easier to identify restricted values and other key aspects, enhancing simplification.

Analyzing both the numerator and denominator of a rational function gives insight into both the structure and potential intercepts.

Numerator Examination

The numerator in our function, \( 2 \), is constant. For simple rational functions, the numerator determines x-intercepts when set equal to zero. Here, since 2 is non-zero, there are no x-intercepts.

Denominator Exploration

The denominator \( x^2 + 3x - 4 \), when factored as \( (x + 4)(x - 1) \), points to values \( x = -4 \) and \( x = 1 \) as not being in the domain, i.e., they make the denominator zero, leading to undefined results.
Understanding these points is crucial because:

• They tell us where vertical asymptotes might occur.
• They clarify interruptions in the graph and function domain.

This investigation requires careful treatment both to avoid division by zero and to appreciate the function's graphical features.