9.2 Key Features of a Parabola

A parabola is the graph of a quadratic function y = ax2 + bx + c There are four key features that we are interested in: Y-Intercept X-Intercept Axis of Symmetry The turning point

Y-Intercept y x 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-Intercept(s) x y 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 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 Hint: Looks like a cup y x 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 Minimum Hint: Looks like a hat y x Maximum

Example: Find the Y-intercept for y = x2 - 6x + 8 y = x2 - 6x + 8 Let x = 0 and substitute in to the equation y = (0)2 - 6(0) + 8 y = 0 - 0 + 8 y = 8 The co-ordinates of the Y-intercept are (0,8) (0,8)

Example: Find the X-intercept for y = x2 - 6x + 8 y = x2 - 6x + 8 Let y = 0 0 = x2 - 6x + 8 Factorise to find x 0 = (x - 2)(x - 4) x = 2 and x = 4 The co-ordinates are (2,0) & (4,0) (0,8) (2,0) (4,0)

Example: Find the axis of symmetry for y = x2 - 6x + 8 Graphically: The X-intercepts are (2,0) & (4,0). Therefore the axis of symmetry is the line x = 3 OR Algebraically: x = -b/2a a = 1 b = -6 c = 8  x = 6/2 x = 3 (0,8) (2,0) (4,0) x =3

Example: Find the turning point for y = x2 - 6x + 8 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 = x2 - 6x + 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) (4,0) (3,-1) x =3

Things to remember about Parabolas Its equation is a quadratic and contains an x2 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