9 2 Key Features of a Parabola A

9. 2 Key Features of a Parabola

A parabola is the graph of a quadratic function y = ax 2 + bx + c • There are four key features that we are interested in: v. Y-Intercept v. X-Intercept v. Axis of Symmetry v. The turning point

Y-Intercept y • The Y-intercept is the point at which the parabola crosses the y -axis • All parabolas have a y -intercept • It can be found when x=0 y-Intercept x

X-Intercept(s) • The X-intercept is the point, or points at which the parabola crosses the x-axis • It can be found when y=0 • Parabolas can have 0, 1 or 2 x-intercepts y x no X-intercepts 2 X-intercepts 1 X-intercept

Axis of Symmetry • The axis of symmetry is a vertical line that divides a parabola into two halves. • It can be found by looking at a graph or by its equation. • Graph: Half way between X-intercepts. • Equation: y x Axis of symmetry

Turning Point • The turning point is the point on the graph that the parabola changes direction • Can either be a minimum or maximum • All parabolas have a minimum or maximum • The x-coordinate of the turning point can be found by finding the axis of symmetry • The y-coordinate turning point can be found by substituting the x-coordinate of the turning point Hint: Looks like a cup y x Minimum Hint: Looks like a hat y x Maximum

Example: Find the Y-intercept for y = x 2 - 6 x + 8 y y = x 2 - 6 x + 8 Let x = 0 and substitute in to the equation y = (0)2 - 6(0) + 8 y=0 -0+8 y=8 (0, 8) x The co-ordinates of the Y -intercept are (0, 8)

Example: Find the X-intercept for y = x 2 - 6 x + 8 y y = x 2 - 6 x + 8 Let y = 0 0 = x 2 - 6 x + 8 Factorise to find x 0 = (x - 2)(x - 4) x = 2 and x = 4 (0, 8) (2, 0) The co-ordinates are (2, 0) & (4, 0) x

Example: Find the axis of symmetry for y = x 2 - 6 x + 8 • Graphically: The X-intercepts are (2, 0) & (4, 0). Therefore the axis of symmetry is the line x = 3 OR • Algebraically: x = -b/2 a a = 1 b = -6 c = 8 x = 6/2 x=3 y (0, 8) (2, 0) (4, 0) x =3 x

Example: Find the turning point for y = x 2 - 6 x + 8 y We already know that the axis of symmetry is x = 3. This means that the X-coordinate of the turning point is 3. To find the y-coordinate substitute 3 into the original equation. y = x 2 - 6 x + 8 Let x = 3 y = (3)2 - 6(3) + 8 y = 9 - 18 + 8 y = -1 The coordinates of the turning point are (3, -1) (0, 8) (2, 0) x (4, 0) (3, -1) x =3

Things to remember about Parabolas • Its equation is a quadratic and contains an x 2 term • Its a smooth curve • It is symmetrical and has an axis of symmetry • There is always a turning point that is either a minimum or a maximum