The factored form of a quadratic function expresses the quadratic as a product of two linear factors.
A quadratic function is typically represented in standard form as:
f(x) = ax² + bx + c,
where a, b, and c are constants and a ≠ 0.
The factored form of the same quadratic function looks like this:
f(x) = a(x - r₁)(x - r₂),
where:
- a is the same leading coefficient as in the standard form.
- r₁ and r₂ are the roots or x-intercepts of the quadratic function (the values of x for which f(x) = 0).
Key Characteristics and Benefits of Factored Form:
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Finding the Roots: The factored form makes it easy to identify the roots (x-intercepts) of the quadratic. By setting each factor equal to zero, you can solve for x and find the roots. For example, in the equation f(x) = a(x - r₁)(x - r₂), the roots are x = r₁ and x = r₂.
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Graphing the Quadratic: Knowing the roots allows you to quickly sketch the graph of the parabola. The parabola intersects the x-axis at x = r₁ and x = r₂.
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Converting to Standard Form: You can convert the factored form back to standard form by expanding (multiplying out) the factors and simplifying. For example, expanding f(x) = a(x - r₁)(x - r₂) gives you back the form f(x) = ax² + bx + c.
Example:
Consider the quadratic function in factored form:
f(x) = 2(x - 3)(x + 1)
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Roots: The roots are x = 3 and x = -1. These are the x-intercepts of the parabola.
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Conversion to Standard Form: Expand the factored form:
f(x) = 2(x² + x - 3x - 3)
f(x) = 2(x² - 2x - 3)
f(x) = 2x² - 4x - 6Now the quadratic function is in standard form: f(x) = 2x² - 4x - 6.
In summary, the factored form of a quadratic function, f(x) = a(x - r₁)(x - r₂), directly reveals the roots of the quadratic, which are crucial for understanding and graphing the function.
