Define the quadratic function ƒ having x-intercepts (1, 0) and (-2, 0) and y-intercept (0, 4).

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Recall that a quadratic function with x-intercepts at points $(r_1, 0)$ and $(r_2, 0)$ can be expressed in factored form as: \[f(x) = a(x - r_1)(x - r_2)\] where $a$ is a constant that affects the vertical stretch or compression of the parabola.

Substitute the given x-intercepts $r_1 = 1$ and $r_2 = -2$ into the factored form: \[f(x) = a(x - 1)(x + 2)\]

Use the y-intercept $(0, 4)$ to find the value of $a$. Substitute $x = 0$ and $f(0) = 4$ into the equation: \[4 = a(0 - 1)(0 + 2)\]

Simplify the right side of the equation to solve for $a$: \[4 = a(-1)(2) = -2a\] Then solve for $a$ by dividing both sides by $-2$: \[a = \frac{4}{-2}\]

Write the final quadratic function by substituting the value of $a$ back into the factored form: \[f(x) = a(x - 1)(x + 2)\]

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Function Standard Form

A quadratic function is typically written as f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. This form helps identify the shape and position of the parabola on the coordinate plane.

Using x-Intercepts to Find Factors

The x-intercepts of a quadratic function correspond to the roots of the equation f(x) = 0. If the roots are r1 and r2, the function can be expressed as f(x) = a(x - r1)(x - r2), which helps in constructing the quadratic from given zeros.

Determining the Leading Coefficient Using the y-Intercept

The y-intercept is the value of f(x) when x = 0, which equals c in the standard form. By substituting x = 0 and the given y-intercept value into the factored form, we can solve for the leading coefficient a to fully define the quadratic function.

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