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Factored form is a way of expressing a quadratic equation that highlights its roots, also known as zeros. Every quadratic function can be written in the form \( y = a(x - p)(x - q) \), where \( p \) and \( q \) are the roots of the polynomial, and \( a \) is a constant that affects the width and direction of the parabola.
For example, to find the factored form given the roots \( x = -1 \) and \( x = 4 \), we write the equation as \( y = a(x + 1)(x - 4) \). The terms in the parentheses \( (x + 1) \) and \( (x - 4) \) correspond to the condition \( x = -1 \) and \( x = 4 \) being zeros.
Factored form is very useful in identifying the zeros of the quadratic function quickly and visually interpreting where the graph crosses the x-axis. This format is a stepping stone to transform the equation into its standard polynomial form by expanding the factors.

The y-intercept of a function is where the graph intersects the y-axis. In simpler terms, it's the value of \( y \) when \( x \) is zero. For quadratic functions, it provides vital information about the position of the curve on the coordinate plane.
For the given quadratic function, the y-intercept is at the point \( (0, 8) \). This tells us that when \( x = 0 \), \( y \) must equal 8.

• To find the y-intercept in the function's equation, substitute \( x = 0 \) in the polynomial.
• In our transformed function \( y = -2(x^2 - 3x - 4) \), replacing \( x \) with 0 gives \( y = 8 \), which matches the stated y-intercept.

By observing the y-intercept, we can discover key points about the function that help us determine the remaining parts of the equation, particularly the scaling factor \( a \).

Expanding polynomials is the process of simplifying expressions, often involving two binomials, into a standard polynomial form. This involves distributing terms and combining like terms to form a quadratic expression in the format \( y = ax^2 + bx + c \).
For the quadratic with zeros at \( x = -1 \) and \( x = 4 \), the factored form \( y = a(x + 1)(x - 4) \) can be expanded:

• First, multiply the binomials \( (x + 1)(x - 4) \) to get \( x^2 - 3x - 4 \).
• Then, involve the constant \( a \) in further simplification.

This step results in the expression \( y = a(x^2 - 3x - 4) \). This expanded form is critical to identify the full quadratic equation, enabling us to determine the precise shape of the parabola by finding the appropriate value of \( a \) using other given points like the y-intercept.