130 Quadratics - Basic Facts GraphOrg

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© Mr Vasquez

QUADRATIC FUNCTION y = ax2 + bx + c (a ≠ 0) Graph: Parabola

y-intercept:

Point where the graph intersects the y-axis. Value of y when x = 0. y = (0, c)

x-intercepts: Points where the graph intersects the x-axis. Values of x when y = 0.  b  b 2  4ac   b  b 2  4ac  x1   , 0 x2   , 0 2a 2a     Axis of Symmetry / Line of Symmetry Straight line with respect to which a figure is symmetric. Vertical line containing the parabola’s minimum (or maximum) point. b Equation: x 2a Vertex: Point that defines the minimum or maximum of the graph. Point of intersection of the parabola and the axis of symmetry. b x-value: x 2a y-value: Substitute the value of x into the equation. Then find y. y = x2 − 2x − 3

Root A root is a solution of an equation. Roots (Solution) of a Quadratic Equation x-coordinates of the points where the graph of the quadratic function crosses the x-axis. Zero of a Function A value of x (the argument of a function) at which y (the value of the function) is zero. For example, y = x2 − 3x has a zero at x = 3.

y = 1x2 − 6x − 7 a is positive (a > 0) The parabola opens upwards. It has a minimum (lowest) point.

Methods for Solving Quadratic Equations 1. Quadratic Formula 1.1 Graphing calculator program based on the Quadratic Formula. 1.2 Paper-and-pencil calculation using the Quadratic Formula. 2. Completing the square 3. Graphing 4. Factoring Quadratic Formula x

b  b2  4ac 2a

Discriminant: b2 − 4ac It is the expression under the radical sign. Discriminant and Nature of Roots If b2 − 4ac > 0, the equation has two real roots. If b2 − 4ac = 0, the equation has one real root. If b2 − 4ac < 0, the equation has no real roots, only imaginary roots. These roots can be written using the imaginary unit i. Imaginary Unit (i)

i 2  1

Therefore, i  1 Zero Product Property For any real numbers a and b, if a(b) = 0, then a = 0 or b = 0. The factoring method depends on the Zero Product Property. y = −1x2 + 2x + 3 a is negative (a < 0) The parabola opens downwards. It has a maximum (highest) point.

Sum and Product of Roots If the roots of ax2 + bx + c = 0, with a ≠ 0, are s1 and s2, then: b c and s1  s2   s1s2  a a