what information can we visually determine from this quadratic graph? vertex vertical stretch factor...

What information can we visually determine from this Quadratic Graph?

–2 –1 1 2 3 4 5 6 7 8 9 10

–4

–3

–2

–1

1

2

3

4

x

y

Vertex

Vertical Stretch Factor

(4, -3)

Over 1, up 1

Over 2, up 4

Normal Pattern

so VS=1

2)()(1

pxqya

2)4()3( xySo the function is

What information can we visually determine from this Quadratic Graph?

Vertex

Vertical Stretch Factor

(-2, 1)

Over 1, up ?

Over 2, up ?

Over 3, up 3 ( )

so VS=

2)()(1

pxqya

2)2()1(3 xySo the function is

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4

5

10

x

y

9x3

1

3

1

What about this Quadratic Graph?

Vertex

Vertical Stretch Factor

(1, -5)

Over 1, down 2

Over 2, down 8

so VS= -2

2)()(1

pxqya

2)1()5(2

1 xySo the function is

)1x2(

–2 –1 1 2 3 4

–20–19–18–17–16–15–14–13–12–11–10–9–8–7–6–5–4–3–2–1

12

xy

)4x2(

Now it’s your turn!Determine the Functions that model each of these curves:

–10 –5 5 10

–10

–8

–6

–4

–2

2

4

6

8

10

x

y

Curve B

Curve A 27x y

2) 2 ( ) 3 ( 2 x y

Determine the Functions that model each of these curves:

Curve B 2) 8 ( ) 3 ( x y

–10 –5 5 10

–10

–8

–6

–4

–2

2

4

6

8

10

x

y

Curve A 2) 5 ( ) 2 ( 2 x y

2 Sad Parabolas

Why?

a < 1

Determine the Functions that model each of these curves:

Curve B

Curve A 2) 4 ( 8 x y

2) 5 ( ) 9 (

5

1 x y

–10 –5 5 10

–10

–8

–6

–4

–2

2

4

6

8

10

x

y

2 Happy Parabolas

Why?

a > 1

Remember, quadratic functions represent real-life situations such as:

Real Life Connection

…the height of a kicked soccer ball

y = (-9.8)t2 + 20t

…the motion of falling objects pulled by gravity:

height = -16t2 + 100

Graphs of Quadratic Functions

Quadratic Functions graph into a shape called a

Parabola

If the vertical stretch, “a”, is negative, the

parabola opens downward

If the vertical stretch, “a”, is positive, the

parabola opens upward

Vertex

Maximum Point

Minimum Point

2)5( xy 2)4( xy

Determining Functions of Parabolas ALGEBRAICALLY!

1. Find the equation of the parabola having a vertex of (3, 5) passing through the point (1, 2).

)5,3(),( qp)2,1(),( yx

2)31()52(1

a

SUBSTITUTE

2)2()3(1

a

4)3(1

a

3

41

a

The FUNCTION for this parabola is:

2)3()5(3

4 xy

Determining Functions of Parabolas ALGEBRAICALLY!

2. Find the equation of the parabola having a vertex of (-1, 4) with a y-intercept of –3.

)4,1(),( qp)3,0(),( yx

2)10()43(1

a

SUBSTITUTE

2)1()7(1

a

1)7(1

a

7

11

a

The FUNCTION for this parabola is:

2)1()4(7

1 xy

Determining Functions of Parabolas ALGEBRAICALLY!

3. A quadratic function has x-intercepts (-1, 0) and (5, 0). Find the formula for the function if it has a maximum value of 6.

)6,2(),( qp)0,5(),( yx

2)25()60(1

a

SUBSTITUTE

2)3()6(1

a

9)6(1

a

2

31

a

The FUNCTION for this parabola is:

2)2()6(2

3 xy

The axis of symmetry is located halfway between the two intercepts so x=2 is the axis of symmetry.

Determining Functions of Parabolas ALGEBRAICALLY!

4. A bridge is shaped as a parabolic arch. If the maximum height is 20 m and the bridge spans 60 m, find the function that models this situation. )20,30(),( qp

)0,0(),( yx

2)300()200(1

a

900)20(1

a

451

a

The FUNCTION for this model is:

2)30()20(45 xy10 20 30 40 50 60

10

20

x

y

20 m

60 m

Homework: Page 38 #39. 42, 43, 44