1. 9.2 Key Features of a Parabola

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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 = y = 8 The co-ordinates of the Y-intercept are (0,8)
(0,8)

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)

9. 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

10. 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 = y = -1 The coordinates of the turning point are (3, -1)
(0,8)
(2,0)
(4,0)
(3,-1)
x =3

11. 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