if the second difference is positive, the graph opens up and the graph opens down if the second...

If the second difference is positive, the graph opens up and the graph opens down if the second difference is negative.

The axis of symmetry is the vertical line which passes through the vertex.

If the coordinates of the vertex is (h, k), the equation of the axis of symmetry is x = h.

(h, k)

(h, k)

x = h

x = h3.2 Properties of Quadratic Relations

y = ax2+bx+c a > 0

y = ax2+bx+c a < 0

The y-coordinate of the vertex is called the optimum value of the relation.

(h, k)

(h, k)

y = k

The optimum value is called a minimum if the parabola opens up and a maximum if the parabola opens down.

The axis of symmetry is the perpendicular bisector of any horizontal line segment joining two points on the parabola.

If the parabola crosses the x-axis, the x-coordinates are called the zeros or x-intercepts.

Determine the following:

a) the coordinates of the vertex

c) the equation of the axis of symmetry.

b) the optimum value

d) the zeros of the relation.(2, – 4)

(2, – 4)

– 4

x = 2

0 and 4

Example 1: Sketch the graph of y = 3x2 + 12x

x y

- 5

- 4

- 3

- 2

-1

0

Start with a table of values.

x y

- 5 15

- 4 0

- 3 - 9

- 2 -12

-1 - 9

0 0

• The zeros are at 0 and – 4.

• Axis of symmetry (halfway between the zeros).

x = – 2

• Substitute x = – 2 into the original equation to obtain.the vertex

2

40 x

y = 3x2 + 12x

y = 3(– 2)2 + 12(– 2)

y = – 12

The vertex is at

(–2 , -12 )

zeros

Axis of symmetry x = – 2

Substitute x = – 2

Ex 2: The following points lie on a parabola. Determine the equation of the axis of symmetry.

a) (3, 2) and (5, 2)(3, 2) (5, 2)

The axis of symmetry lies halfway between 3 and 5.

2

53x

The following points lie on a parabola. Determine the equation of the axis of symmetry.

b) (–3.25, –2) and (2.5, – 2)

The axis of symmetry lies halfway between –3.25 and 2.5.

(2.5, –2)(–3.25, –2)

3.25 2.5

2

The equation of the axis of symmetry is x = – 0.375.

= – 0.375

A golf ball is hit in the air. Its height is given by the equation: h = 50t – 5t2, where h is the height in metres and t is the time in seconds.

b) When does the ball hit the ground? c) What are the coordinates of the vertex? d) Graph the relation. e) What is the maximum height of the golf ball? f) After how many seconds does that occur?

a) What are the zeros of the relation?

Properties of Quadratic Relations (2)

Step 1: Set the WINDOW to thefollowing settings.

Press WINDOW

Reminder (–)

a) What are the zeros of the relation?

Press and enter the equationY= Press GRAPH

Press 2nd TRACE Use arrows to cursor to the left and right of the two x-intercepts (or zeros).

C:\Documents and Settings\Cheryl Ann\My Documents\MPM 2D1\Unit 3\Golf Example.84state

b) When does the ball hit the ground? The ball hits the ground at 10 seconds.

a) What are the zeros of the relation?

The zeros are 0 and 10.

d) Graph the relation

e) What is the maximum height of the golf ball?

f) After how many seconds does that occur?

125 m

5 s zeros

vertexc) What are the coordinates of the vertex?

V(5, 125)

Homework: pg 145 #1 – 7, 9 – 15 (for 12 – 15, graph using TOV)