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Published byGiles O’Connor’ Modified over 4 years ago

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Sketching quadratic functions To sketch a quadratic function we need to identify where possible: The y intercept (0, c) The roots by solving ax 2 + bx + c = 0 The axis of symmetry (mid way between the roots) The coordinates of the turning point. The shape:

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The shape The coefficient of x 2 is -1 so the shape is The Y intercept (0, 5) The roots (-5, 0) (1, 0) The axis of symmetry Mid way between -5 and 1 is -2 x = -2 The coordinates of the turning point (-2, 9)

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Completing the square The coordinates of the turning point of a quadratic can also be found by completing the square. This is particularly useful for parabolas that do not cut the x – axis. REMEMBER

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Axis of symmetry is x = 2 Coordinates of the minimum turning point is (2, 1)

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Axis of symmetry is x = 3 Coordinates of the maximum turning point is (3, 16)

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Solving quadratic equations Quadratic equations may be solved by: The Graph Factorising Completing the square Using the quadratic formula

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This does not factorise.

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Quadratic inequations A quadratic inequation can be solved by using a sketch of the quadratic function. First do a quick sketch of the graph of the function. Roots are -4 and 1.5 Shape is a -4 1.5 The function is positive when it is above the x axis.

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First do a quick sketch of the graph of the function. Roots are -4 and 1.5 Shape is a -4 1.5 The function is negative when it is below the x axis.

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The quadratic formula

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From the above example when the number under the square root sign is zero there is only 1 solution.

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From the above example we require the number under the square root sign to be positive in order for 2 real roots to exist.

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This leads to the following observation. Since the discriminant is zero, the roots are real and equal.

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Using the discriminant We can use the discriminant to find unknown coefficients in a quadratic equation.

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Since the discriminant is always greater than or equal to zero, the roots of the equation are always real.

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Conditions for tangency To determine whether a straight line cuts, touches or does not meet a curve the equation of the line is substituted into the equation of the curve. When a quadratic equation results, the discriminant can be used to find the number of points of intersection.

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Since the discriminant is zero, the line is a tangent to the curve. Hence the point of intersection is (1, 1).

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Hence the equation of the tangent is y = 2x.

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Hence the equation of the two tangents are y = 8x – 2 and y = -8x - 2.

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