Quadratic Functions

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What is the general form of a quadratic function?
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The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.
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What is the shape of the graph of a quadratic function?
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The graph of a quadratic function is a parabola.
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What determines the direction in which a parabola opens?
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The sign of the leading coefficient 'a'. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.
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What is the axis of symmetry in a quadratic function?
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The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. Its equation is x = -b/(2a).
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Where is the vertex of a parabola located?
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The vertex is located at the point (h, k). For a quadratic function in standard form, it is at (-b/(2a), f(-b/(2a))).
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What are the roots of a quadratic function?
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The roots are the x-values where the graph intersects the x-axis. They are also called solutions or zeros of the function.
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How can the roots of a quadratic function be calculated?
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Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
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What is the discriminant in a quadratic function?
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The discriminant is the part of the quadratic formula under the square root: b^2 - 4ac. It determines the number of real roots.
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What does it mean if the discriminant is positive?
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If the discriminant is positive, the quadratic function has two distinct real roots.
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What if the discriminant is zero?
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If the discriminant is zero, there is exactly one real root, meaning the parabola touches the x-axis at one point.
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And if the discriminant is negative?
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If the discriminant is negative, there are no real roots, implying the parabola does not intersect the x-axis.
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Can quadratic functions be solved by factoring?
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Yes, if the quadratic can be factored into a product of binomials, it can be solved using the zero-product property.
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What method can be used besides factoring and the quadratic formula to solve quadratic functions?
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Completing the square is another method used for solving quadratic functions.
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What is the vertex form of a quadratic function?
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The vertex form is f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.
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How does the quadratic function relate to real-world applications?
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Quadratic functions model many real-world situations, such as projectile motion, area problems, and maximum/minimum profit calculations.
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