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The y-intercept of a graph is the point where the graph intersects the y-axis. To find this point, you set the value of \( x \) to zero in the equation. This is because any point on the y-axis has an x-coordinate of zero.

For our equation, \( y = \frac{x^2 - x - 20}{x + 6} \), when \( x = 0 \), the equation becomes:

• \( y = \frac{0^2 - 0 - 20}{0 + 6} \)
• \( y = \frac{-20}{6} \)
• \( y = -\frac{10}{3} \)

Thus, the y-intercept is \( (0, -\frac{10}{3}) \). This is the value of the function at the point where the graph crosses the y-axis. Knowing how to find this intercept helps in sketching the graph and understanding the behavior of the rational function.

The x-intercepts are the points where the graph intersects the x-axis. At these points, the value of \( y \) is zero. To find them, set the output or \( y \) equal to zero and solve the equation for \( x \).

In our function \( y = \frac{x^2 - x - 20}{x + 6} \), setting \( y = 0 \) implies:

• \( \frac{x^2 - x - 20}{x + 6} = 0 \)
• The numerator must be zero: \( x^2 - x - 20 = 0 \)
• Factoring gives \((x - 5)(x + 4) = 0\)
• Thus, \( x = 5 \) and \( x = -4 \).

Therefore, the x-intercepts are \( (5, 0) \) and \( (-4, 0) \). These solutions give us the points on the graph where it crosses the x-axis. The x-intercepts can provide insight into the roots of the equation and are essential for understanding the graph's layout.

Symmetry in graphs can occur with respect to the x-axis, y-axis, or the origin. Testing for symmetry helps us understand the shape and structure of the graph without actually plotting it.

• To check symmetry about the x-axis, replace \( y \) with \( -y \). If the equation remains unchanged, the graph is symmetric with respect to the x-axis. Our function gives \( -y = \frac{x^2 - x - 20}{x + 6} \), which is not equal to the original equation, so there is no symmetry about the x-axis.
• For symmetry about the y-axis, replace \( x \) with \( -x \). If it remains the same, then symmetry exists. For our equation, it results in \( y = \frac{x^2 + x - 20}{-x + 6} \), which is different, indicating no symmetry about the y-axis.
• For symmetry about the origin, replace both \( x \) and \( y \) with \( -x \) and \( -y \) respectively. The equation becomes \( -y = \frac{x^2 + x - 20}{-x + 6} \), which also differs from the original, showing no origin symmetry.

Understanding symmetry simplifies graphing and gives clues about the behavior of rational functions.

Quadratic equations are of the form \( ax^2 + bx + c = 0 \) and can often be solved by factoring. In other cases, the quadratic formula or completing the square might be necessary. Here, we needed to find x-intercepts by solving a quadratic equation. Let's break it down.

The quadratic equation from our example is \( x^2 - x - 20 = 0 \). To solve it:

• First, attempt to factor it into two binomials.
• Notice that \((x - 5)(x + 4) = 0\), meaning the root solutions are \(x = 5\) and \(x = -4\).

Factoring involves finding two numbers that multiply to \( -20 \) (the last term) and add to \( -1 \) (the middle coefficient). In this case, \(-5\) and \(4\) are the numbers that meet these criteria.
Understanding how to handle quadratic equations is fundamental for finding intercepts and analyzing the behavior of polynomials within rational functions. It's a key skill in algebra that enriches your problem-solving toolkit.