What you’ll learn

To graph quadratic functions of the form To identify the axis of symmetry, vertex, domain.range and intercept(s) for a given parabola.To graph quadratic functions of the form

caxy and axy 22

cbxaxy 2

VocabularyQuadratic function, Standard form,Parent function, parabola, axis of symmetry, vertex, minimum, maximum

Quadratic Graphs and Their Properties. Quadratic Functions.

Take a Note: A quadratic function is a function thatcan be written in the form This form is called the standard form of a quadraticfunction.

0a where ,cbxaxy 2

Examples: 2xxy ,9xy ,xy 222 The simplest quadratic function is the quadratic parent function.

22 xy ,xxf

The graph od the quadratic function is called parabola and has a U shape. You can fold a parabola so that the two sides match exactly. The fold or line that divides the parabola into matching halves is called theaxis of symmetry.

Problem 1: Graphing a Quadratic. Identifying a vertex.

Graph.Step 1: Make a table.

X-2 4-1 10 01 12 43 9

2xy 2x2y

3x2

1y)c

x2y)b

xy)a

2

2

2

3x

2

1y 2

5322

1 2

5.3312

1 2

3302

1 2

5.3312

1 2

5322

1 2

5.7332

1 2

822 2

212 2

002 2

822 2

212 2

1832 2

Step 2: Plot the pointsa)(-2, 4) (-1,1) (0,0) (1,1) (2,4)

b)(-2,-8) (-1,-2) (0,0) (1,-2) (2,-8) c)(-2,5) (-1,3.5) (0,3) (1,3.5) (2,5)

minimum up opens

)0,0(vertex

y,0y/yR

x/xD

maximum down opens

)0,0(:vertex

y,0y/yR

x/x:D

3in minimum up opens

)3,0(vertex

y,3y/yR

x/xD

Take a note: The highest or the lowest point of aparabola is a vertex, which is on the axis of

symmetry

cbxaxy in 0a If 2

upward. opensparabola the

parabola.the of point, lowestthe or

point minimumthe isvertex The

downward. opensparabola the

cbxaxy in0 a If 2

parabola.the of point, highestthe or

point maximumthe isvertex The

Your turn:A child drops a pebble from a height of 30 ft above a lake. The function gives the height h of the pebble(in feet) after t seconds. What is the graph of this quadraticfunction? At about what time does the pebble hit the water?

30t16y 2

t37.1

t875.1

t16

30

t1630

30t16y

2

2

2

2

Take a note: There is another way to graph a parabola . The graph of , where has the line as its axis of as symmetry . The x-coordinate of the vertex is and the

y-coordinate of the vertex is . You can

use the coordinates of the vertex, the axis of symmetry and the y-intercept (meaning when x=0)to help you graph a quadratic function.

cbxaxy 2

a2

b

cbxaxy 2

a2

bfy

a2

b

Problem 3: What is the graph of ?

4x6xy 2

Step 1: Identify a, b, and c. 4c -6b 1a Step 2: find the vertex, remember the x-value is the axis of symmetry.

312

6

2a

b-xvertex

543633fy 2 )5,3(

Step 3: The y-intercept (0,4) 4y

4060y 2

Step 4: Find another point in the side as the vertex.

)1,1(

14161y

1x2

Your turn: Graph 2x4xy 2

(0,-2) -2y then0 x when

)2,2(

22242f(-2)

2xvertex the for Now

212

4

2a

bsymmetry - of axis

2

Answer

Problem 4: A baseball is thrown into the air with an upward velocity of 30ft/s. Its height h, in feet, after t seconds is given by the function .How long will take the ball to reach its maximum height? What is the ball’s maximum height? What is the range of the function?

6t30t16y 2

sec16

15

162

30

a2

b

16

120h0

ft16

1206

16

1530

16

15

16

15f

2

Answer

Your turn: During a halftime basketball game, a slingshot launches T-shirts at the crowd. A T-shirt is launched with anInitial velocity of 72ft/sec. The T-shirt is caught 35 ft. above the court. How long will take the T-shirt to reach itsmaximum height? What is the maximum height? What is therange of the function that models the height of the T-shirt over time?

86h5

86525.27225.216h

25.2162

72

a2

bt

2

Answer

Classwork odd Homework evenTB pgs. 538-539 exercises. 7-48 pgs. 544-545 exercises 7-33