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X-intercepts are a crucial part of understanding the behavior of a graph. They represent the points where a graph crosses the x-axis. In simpler terms, these are the values of \( x \) where the function equals zero. For rational functions, x-intercepts are found by setting the numerator equal to zero.

In the given function, \( t(x) = \frac{x^2 - x - 2}{x-6} \), we focus on the numerator \( x^2 - x - 2 \). To find the x-intercepts, solve \( x^2 - x - 2 = 0 \). Once factored, it becomes \( (x-2)(x+1) = 0 \).

The solutions \( x = 2 \) and \( x = -1 \) indicate the x-intercepts. This means that on the x-axis, the graph crosses or touches the x-axis at these points.

Y-intercepts identify the points where the graph of a function crosses the y-axis. Essentially, this is where the value of \( x \) equals zero.

Finding the y-intercept for a rational function like \( t(x) = \frac{x^2 - x - 2}{x-6} \) is straightforward. Substitute \( x = 0 \) into the function.

After calculation: \( t(0) = \frac{0^2 - 0 - 2}{0-6} = \frac{-2}{-6} = \frac{1}{3} \).

Thus, the graph intersects the y-axis at \( \left( 0, \frac{1}{3} \right) \). This point helps to anchor the graph on the coordinate plane.

Quadratic equations often appear when dealing with the numerator of a rational function. These are equations in the form \( ax^2 + bx + c = 0 \). Solving a quadratic equation can provide x-intercepts of the function.

In this exercise, the numerator \( x^2 - x - 2 \) forms a quadratic equation. Solving this involves either factoring or applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Both methods aim to find the roots or solutions for \( x \). Here, we chose to factor: \( (x-2)(x+1) = 0 \). The roots \( x = 2 \) and \( x = -1 \) emerge, representing where the graph touches the x-axis.

Factoring is a technique used to simplify expressions and solve equations more easily. It transforms complex expressions into more manageable products of simpler expressions.

For the quadratic equation in our exercise \( x^2 - x - 2 \), we factor it into \( (x-2)(x+1) \). This conversion transforms a complex polynomial into a simple product, making it easier to find solutions.

Evaluate each factor to zero separately to get the intercepts:

• If \( x-2 = 0 \), then \( x=2 \)
• If \( x+1 = 0 \), then \( x=-1 \)

Hence, factoring serves as a critical step in solving for x-intercepts of a rational function.

Rational expressions are fractions made from polynomials. They follow the structure \( \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials. These expressions are prevalent in algebra and calculus.

In the problem, \( t(x) = \frac{x^2 - x - 2}{x-6} \) is our rational function. The numerator is \( x^2 - x - 2 \), and the denominator is \( x-6 \). Simplifying these expressions can provide valuable insights into their behavior, such as identifying intercepts and asymptotes.

Remember not to set a rational expression in its entirety to zero immediately. Instead, focus on the numerator to find zeros, like we did with the x-intercepts. In this process, rational expressions can be quite versatile in representing intricate behaviors of mathematical functions.