Quadratic Functions in Factored Form

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What is a quadratic function in factored form?

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A quadratic function in factored form is expressed as f(x) = a(x - r1)(x - r2), where 'a' is a non-zero constant, and r1 and r2 are the roots of the equation.

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What are the x-intercepts of a quadratic function?

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The x-intercepts of a quadratic function are the points where the graph intersects the x-axis, found by setting f(x) = 0 and solving for x.

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How do you find the y-intercept of a quadratic function?

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The y-intercept of a quadratic function is found by evaluating f(0), which gives the value of the function when x = 0.

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What does the 'a' in the quadratic function f(x) = a(x - r1)(x - r2) represent?

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The 'a' represents the leading coefficient, which affects the width and direction of the parabola (upward if a > 0, downward if a < 0).

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What is the vertex of a parabola in factored form?

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The vertex of a parabola can be found using the formula x = (r1 + r2)/2, where r1 and r2 are the roots of the quadratic function.

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How do you determine the direction of a parabola?

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The direction of a parabola is determined by the sign of the leading coefficient 'a': if a > 0, the parabola opens upwards; if a < 0, it opens downwards.

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What is the significance of the roots in a quadratic function?

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The roots (or zeros) of a quadratic function are the values of x that make the function equal to zero, indicating where the graph intersects the x-axis.

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How can you convert a quadratic function from standard form to factored form?

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To convert from standard form (ax^2 + bx + c) to factored form, you can use factoring techniques or the quadratic formula to find the roots.

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What is the relationship between the coefficients and the roots of a quadratic function?

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For a quadratic function f(x) = ax^2 + bx + c, the sum of the roots is -b/a and the product of the roots is c/a.

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How do you find the maximum or minimum value of a quadratic function?

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The maximum or minimum value occurs at the vertex, which can be found using x = -b/(2a) in the standard form of the quadratic equation.