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

검증된 단계별 안내

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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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.