30

Section 2.3 – Quadratic Functions and Models

Quadratic Function

A function f is a quadratic function if2( )f x ax bx c

Vertex of a Parabola 2( ) 4 2f x x x

The vertex of the graph of ( )f x is

2

or bx v a

V x

2 or b

y v aV y f

2 2 Point b b

a aVertex f ,

42 2(1)

2bxa

(2)2by f fa

22 4(2) 2

6

Vertex point: 2, 6

Axis of Symmetry: 2bx Vx a

Axis of Symmetry: x = 2

Minimum or Maximum Point

If a > 0 f(x) has a minimum point

If a < 0 f(x) has a maximum point

@ vertex point ,x yV V

Minimum point @ 2, 6

Range

If a > 0 , yV

If a < 0 , yV

6,

Domain: ,

x

y

Minimum / Vertex point

Symmetry Line

x-intercept

2( ) 4 2f x x x

31

Example

For the graph of the function 2( ) 6 3f x x x

a. Find the vertex point

62(1)

3x

2( 3) ( 3) 6( 3) 3 6y f

Vertex point (3, 6)

b. Find the line of symmetry: x = 3

c. State whether there is a maximum or minimum value and find that value

Minimum point, value (3, 6)

d. Find the x-intercept

26 6 4(1)(3) 6 24 6 2 6

2(1) 2 2

3 6

3 6

3 60.5

5.45x

e. Find the y-intercept

y = 3

f. Find the range and the domain of the function.

Range: [6, ∞)

Domain: (∞, ∞)

g. Graph the function and label, show part a thru d on the plot below

h. On what intervals is the function increasing? Decreasing?

Decreasing: (∞,3)

Increasing: (3, ∞)

x

y

Symmetry: x = 3

Vertex Point / Min (3, 6)

x-intercept

2( ) 6 3f x x x

32

Example

Find the axis and vertex of the parabola having equation 2( ) 2 4 5f x x x

Solution

2bxa

42(2)

1

Axis of the parabola: 1x

( 1)y f

22( 1) 4( 1) 5

3

Vertex point: 1,3

Maximizing Area

You have 120 ft of fencing to enclose a rectangular region. Find the dimensions of the rectangle that

maximize the enclosed area. What is the maximum area?

Solution

wlP 22

wl 22120

wl 60 60l w

A lw

(60 )w w

260w w

2 60w w

Vertex: 30)1(2

60

w

3060 wl

2(30)(30 90) 0 ftA lw

33

Example

A stone mason has enough stones to enclose a

rectangular patio with 60 ft of stone wall. If the house

forms one side of the rectangle, what is the maximum

area that the mason can enclose? What should the

dimensions of the patio be in order to yield this area?

Solution

2 60P l w 60 2l w

A lw (60 2 )ww

260 2w w 22 60w w

2ba

w

602( 2)

15 ft

60 2 60 2(15) 30l w ft

Area = (15)(30) = 450 ft2

Position Function (Projectile Motion)

Example

A model rocket is launched with an initial velocity of 100 ft/sec from the top of a hill that is 20 ft high. Its

height t seconds after it has been launched is given by the function 2( ) 16 100 20s t t t . Determine

the time at which the rocket reaches its maximum height and find the maximum height.

Solution

2bta

1002( 16)

3.125 sec

3.122

5( ) 16( ) 13.125 3.1200( ) 2 176.20 5 5 s t ft

34

Exercises Section 2.3 – Quadratic Functions and Models

1. Give the vertex, axis, domain, and range. Then, graph the function 2( ) 6 5f x x x

2. Give the vertex, axis, domain, and range. Then, graph the function 2( ) 6 5f x x x

3. Give the vertex, axis, domain, and range. Then, graph the function 2 4 2f x x x

4. Give the vertex, axis, domain, and range. Then, graph the function 22 16 26f x x x

5. You have 600 ft. of fencing to enclose a rectangular plot that borders on a river. If you do not fence

the side along the river, find the length and width of the plot that will maximize the area. What is the

largest area that can be enclosed?

6. A picture frame measures 28 cm by 32 cm and is of uniform width. What is the width of the frame if

192 cm2 of the picture shows?

7. An open box is made from a 10-cm by 20-cm of tin by cutting a square from each corner and

folding up the edges. The area of the resulting base is 96 cm2. What is the length of the sides of the

squares?

35

8. A fourth-grade class decides to enclose a rectangular garden, using the side of the school as one side

of the rectangle. What is the maximum area that the class can enclose with 32 ft. of fence? What

should the dimensions of the garden be in order to yield this area?

9. A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If a

river forms one side of the corrals and 240 yd of fencing is available, what is the largest total area

that can be enclosed?

10. A Norman window is a rectangle with a semicircle on top. Sky Blue Windows is designing a

Norman window that will require 24 ft of trim on the outer edges. What dimensions will allow the

maximum amount of light to enter a house?

36

11. A frog leaps from a stump 3.5 ft. high and lands 3.5 ft. from the base of the stump.

It is determined that the height of the frog as a function of its distance, x, from the base of the stump

is given by the function 20.5 0.75 3.5h x x x where h is in feet.

a) How high is the frog when its horizontal distance from the base of the stump is 2 ft.?

b) At what two distances from the base of the stump after is jumped was the frog 3.6 ft. above the

ground?

c) At what distance from the base did the frog reach its highest point?

d) What was the maximum height reached by the frog?

40

x

y

Solution Section 2.3 – Quadratic Functions

Exercise

Give the vertex, axis, domain, and range. Then, graph the function 2( ) 6 5f x x x

Solution

Vertex: 2ba

x

62(1)

3

23 3( ) ( 6( )3) 5y f

4

Vertex point: 3, 4

Axis of symmetry: 3x

Domain: ,

Range: 4,

Exercise

Give the vertex, axis, domain, and range. Then, graph the function 2( ) 6 5f x x x

Solution

Vertex: 2ba

x

62( 1)

3

23 3( ) ( ) 6( ) 53y f

4

Vertex point: 3, 4

Axis of symmetry: 3x

Domain: ,

Range: , 4

x

y

41

Exercise

Graph the quadratic. Give the vertex, axis of symmetry, domain, and range:

2 4 2f x x x

Solution

Vertex point:

4 2

2 2 1bx a

22 2 4 2 2 2f

The vertex point: 2, 2

Axis of symmetry is: 2x

Domain: ,

Range: 2, (Since function has a minimum)

To graph: find another point:

0 0 2x y f

Exercise

Graph the quadratic. Give the vertex, axis of symmetry, domain, and range:

22 16 26f x x x

Solution

Vertex point:

16 4

2 2 2bx a

24 2 4 16 4 26 6f

The vertex point: 4, 6

Axis of symmetry is: 4x

Domain: ,

Range: , 6 (Since function has a maximum)

To graph: find another point:

3 3 4x y f

x

y

x

y

42

Exercise

You have 600 ft of fencing to enclose a rectangular plot that borders on a river. If you do not fence the

side along the river, find the length and width of the plot that will maximize the area. What is the largest

area that can be enclosed?

Solution

2P l w

600 2l w 600 2l w

A lw

(600 2 )w w

2600 2w w

22 600w w

Vertex: 6002( 2)

150w

600 2 300l w

(300)(150)A lw

245000 ft

Exercise

A picture frame measures 28 cm by 32 cm and is of uniform width. What is the width of the frame if 192

cm2 of the picture shows?

Solution

Area of the picture = 192)228)(232( xx

19245664896 2 xxx

01924120896 2 xx

07041204 2 xx

0176302 xx

0)22)(8( xx

8 0

22 0

8

22

x

x

x

x

43

Exercise

An open box is made from a 10-cm by 20-cm of tin by cutting a square from each corner and folding up

the edges. The area of the resulting base is 96 cm2. What is the length of the sides of the squares?

Solution

Area of the base 20 2 10 2 96x x

2200 40 20 4 96x x x

24 60 200 96 0x x

24 60 104 0x x Solve for x

132, x

The length of the sides of the squares is 3-cm

Exercise

A fourth-grade class decides to enclose a rectangular garden, using the side of the school as one side of

the rectangle. What is the maximum area that the class can enclose with 32 ft. of fence? What should the

dimensions of the garden be in order to yield this area?

Solution

Perimeter: 2 32P l w

32 2l w

Area: A lw

(32 2 )A w w

232 2w w

22 32w w

Vertex: 322( 2)

8w

32 62 18l

16 8A lw

2128 ft

44

Exercise

A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If a river

forms one side of the corrals and 240 yd of fencing is available, what is the largest total area that can be

enclosed?

Solution

Perimeter: 3 240P l w

240 3l w

Area: A lw

(240 3 )A w w

2240 3w w

23 240w w

Vertex: 2402( 3)

40w

240 3 40 120l

120 40A lw

24800 yd

Exercise

A Norman window is a rectangle with a semicircle on top. Sky Blue Windows is designing a Norman

window that will require 24 ft. of trim on the outer edges. What dimensions will allow the maximum

amount of light to enter a house?

Solution

Perimeter of the semi-circle 12

2 x

Perimeter of the rectangle 2 2x y

Total perimeter: 2 2 24x x y

2 24 2y x x

122

y x x

Area 212

2x x y

2 2 122 2

x x x x

2 2 224 2

2x x x x

224 22

x x

45

22 242

x x

2

24 242 2 2 2

24

2

44

ba

x

24 24122 4 4

y

24 96 24 482 4

244

Exercise

A frog leaps from a stump 3.5 ft. high and lands 3.5 ft. from the base of the stump.

It is determined that the height of the frog as a function of its distance, x, from the base of the stump is

given by the function 20.5 0.75 3.5h x x x where h is in feet.

a) How high is the frog when its horizontal distance from the base of the stump is 2 ft.?

b) At what two distances from the base of the stump after is jumped was the frog 3.6 ft. above the

ground?

c) At what distance from the base did the frog reach its highest point?

d) What was the maximum height reached by the frog?

Solution

a) At 2 . 2x ft Find h x

22 0.5 2 0.75 2 3.5 3h ft

b) 20.5 0. 33 .675 .5h x x x

20.5 0. 3.675 3.5 0x x

20.5 0.75 .1 0x x

Solve for x: 0.1, 1.4x ft

c) The distance from the base for the frog to reach the highest point is

.75

2 2 .5.75 ftbx

a

d) Maximum height:

20.5 0.7.75 .75 .75 3.5 3 785 .h x ft

3.6h

2x

46

Exercise

For the graph of the function 2( ) 6 3f x x x

a. Find the vertex point

62(1)

3x

2( 3) ( 3) 6( 3) 3 6y f Vertex point (3, 6)

b. Find the line of symmetry: x = 3

c. State whether there is a maximum or minimum value and find that value

Minimum point, value (3, 6)

d. Find the zeros of f(x)

26 6 4(1)(3) 6 36 12 6 24 6 2 6

2(1) 2 2 23 6x

3 6

3 6

0.5

5.45x

e. Find the y-intercept 3y

f. Find the range and the domain of the function.

Range: [6, ∞) Domain: (∞, ∞)

g. Graph the function and label, show part a thru d on the plot below:

h. On what intervals is the function increasing? Decreasing?

Decreasing: (∞, 3)

Increasing: (3, ∞)

x

y

Symmetry: x = 3

Vertex Point / Min (3, 6)

2( ) 6 3f x x x

Zero’s