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**Characteristics of Quadratics**

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Parabola This graph is created by a 2nd degree equation, called a quadratic. Standard form for a quadratic equation is f(x) = ax2 + bx + c. The U-shape of the graph is called a parabola. The graph continues both left and right, so the domain is all real numbers or R. f(x) = x2 – 5x – 2

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Coefficient of x2 The coefficient of x2 determines the direction in which the parabola opens. For positive coefficients, the parabola opens up. For negative coefficients, the parabola opens down. _ _ g(x) = – x2 + 8x – 12 + + f(x) = x2 + 8x + 13

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Vertex Every parabola has an extreme point where the curve turns back or changes direction. This point is called the vertex. For parabolas that open up, the vertex is a minimum, the lowest y-value on the graph. The range is y ≥ min. For parabolas that open down, the vertex is a maximum, the highest y-value on the graph. The range is y ≤ max. maximum f(x) = x2 + 8x + 13 g(x) = – x2 + 8x – 12 Range: y ≥ –3 Range: y ≤ 4 minimum

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Axis of Symmetry A parabola is symmetrical. The left half is the mirror image of the right half. There is an imaginary vertical line that runs through the vertex and acts as a mirror for the two halves of the parabola. This line is called the axis of symmetry. The equation of the axis of symmetry is always x = x-coordinate of vertex. vertex (2, 7) axis of symmetry x = 2

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Roots The point at which a parabola crosses the x-axis is called an x-intercept, root, or zero (the value of y=0). Parabolas can have zero roots, one root, or two roots, depending on how many times they cross the x-axis. No roots 2 roots 1 root

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y-intercepts The point at which a parabola crosses the y-axis is called a y-intercept. h(x) = – x2 + 5x – 3 The y-intercept may be found by looking at the constant in the standard form of the quadratic equation, f(x) = ax2 + bx + c. The y-intercept is always at (0, c). y-intercept (0, –3)

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