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:

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)

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

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

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

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

This does not factorise.

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 The function is positive when it is above the x axis.

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

The quadratic formula

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

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

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

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

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

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.

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

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

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