faculty.pingry.orgfaculty.pingry.org/bpoprik/documents/Packet1Quadratics.pdf · Alg 3 4 Quadratic Equations and Parabolas The Quadratic Formula Standard Form: y ax bx c= + +2 2 4

Alg 3 1

Quadratic Equations and Parabolas

Completing the Square

A Perfect Square Trinomials

B. Creating perfect squares.

( )

2

2

2

1. 6 10 0

6 ______ 10 _____

________

x x

x x

+ + =

+ + = − +

=

22. 3 4 0x x+ − =

C. Working with coefficients

( )

( )

2

2

2

2

3. 2 16 11 0

2( 8 ) 11

2 8 ____________

____________

x x

x x

x x

− + =

− = −

− + =

=

24. 3 4 5 0x x+ + =

What is the relationship between the

red term and the blue term?

± → ± +

± → ± +

± → ± +

± → ± +

± → ± +

2 2

2 2

2 2

2 2

2 2

(x 1) x x

(x 2) x x

(x 3) x x

(x 4) x

1

4

9

x

(x 5) x

2

4

6

8

1

1

0x

6

25

Alg 3 2


Algebra 3 Assignment # 1

Solve each of the following please.

(1) 212x 31x + 20 = 0− (2)

2x 6x = 0−

(3) 2x 10x 11 = 13− − (4)

212x + 8x 15 = 0−

(5) 2x + 2x 1 = 0− (6)

23x + 4x + 5 = 0

(7) 3 22x + 3x 50x 75 = 0− − (8)

2x + 3 x + 2 =

5x 2 2x 3− −

Alg 3 3


Answers

(1) 5 4 ,

4 3 (2) 0 , 6

(3) 2 , 12− (4) 3 5 ,

2 6−

(5) 1 2− ± (6) 2 i 11

3 3− ±

(7) 3 , 5 , 5

2− − (8) 4 11− ±

Alg 3 4


The Quadratic Formula Standard Form:

2y ax bx c= + + 2 4

2

b b acx

a

− ± −=

The Quadratic Formula finds the Roots of the Function

2 4 5

1, 4, 5

y x x

a b c

= + −

= = = −

2 12

1, 1, 12

y x x

a b c

= − −

= = − = −

2 6 6

_____, _____, _____

y x x

a b c

= + +

= = = x = _________________

Alg 3 5


2 6 12

_____, _____, _____

y x x

a b c

= + +

= = = x = _________________

When we solve ( ) 0f x = , what are we actually solving for graphically?

2 4b ac± − is called the discriminant. It helps us understand where the roots are graphically.

Alg 3 6


Algebra 3 Assignment # 2

Solve each of the following please.

(1) 0 4 5x x6 2 =−+ (2) 0 1 x x2 =+−

(3) 0 4 8x x2 =−− (4) 0 4 x x3 2 =++

(5) 0 2 4x x3 2 =++ (6) 0 18 6x x2 =−+

(7) 0 209 8x x2 =−− (8) ( ) 5 3x 1 x22

−=−

Alg 3 7


Answers

(1) 3

4 ,

2

1− (2)

1 i 3

2 2±

(3) 5 2 4 ± (4) 1 i 47

6 6− ±

(5) 2 i 2

3 3− ± (6) 3 3 3 ±−

(7) 19 , −11 (8) 7 i 47

8 8±

Alg 3 8


Graphing Quadratic Functions

2y x=

Observations:

2 3y x= + 2( 2)y x= + 22y x= 2y x= −

2 1y x= − 2( 3)y x= − 21

4y x=

Alg 3 9


The Parabolic Equation

2f(x) = (x - h)a + k ( )

22 3y x= − +

vertex = (h , k) vertex (2,3) axis of symmetry x = h axis x = 2 a controls shape (steepness) of curve Sign of a controls direction of the concavity Graph 2 points on either side of vertex

21

4=y x

23 5= −y x

x y

0

1

2 3

3

4

(2,3)

x = 2

Alg 3 10


= − − +21. f (x) (x 1) 4

= +22. x y 2

Alg 3 11


Homework Assignment #3 – Graphing Parabolas Graph the following quadratic functions. For each, state the (a) coordinates of the vertex, (b) equation of the axis of symmetry, (c) the domain, and (d) the range.

1. 2( ) ( 3) 2= − −f x x 2. 2( ) ( 3) 2= − + +f x x

Vertex = ( , ) Vertex = ( , )

A.O. S. = A.O. S. = Domain= Domain = Range= Range =

3. 2( ) 2( 1) 3= + +f x x 4. 21( ) ( 1) 4

4= − − +f x x


A.O. S. = A.O. S. = Domain= Domain = Range= Range =

Alg 3 12


5. 2( ) ( 4)= − +f x x 6. 2( ) 8 16= + +f x x x


A.O. S. = A.O. S. = Domain= Domain = Range= Range = Answers 1. Vertex: (3,-2) Axis: x = 3 Domain: All reals Range: y ≥ -2 2. Vertex: (-3,2) Axis: x = -3 Domain: All reals Range: y ≤ 2 3. Vertex: (-1,3) Axis: x = -1 Domain: All reals Range: y ≥ 3 4. Vertex: (1,4) Axis: x = 1 Domain: All reals Range: y ≤ 4 5. Vertex: (-4,0) Axis: x = -4 Domain: All reals Range: y ≤ 0 6. Vertex: (-4,0) Axis: x = -4 Domain: All reals Range: y ≥ 0

Alg 3 13


Back to Completing the Square

Graph 2 10 22= + +y x x

Did we find the vertex? How would we know?

How about graphing 24 4 3= + −y x x

Can we convert this to the 2f(x) = (x - h)a + k format instead?

Converting 2= + +y ax bx c -> 2( )= − +y a x h k

x y

Alg 3 14


Using Quadratic Equations to Solve Problems 1. The sum of a positive number and its square is 72. Find the number. 2. The length of a rectangle is 3 centimeters greater than its width. The area is 70 square centimeters.

Find the dimensions of the rectangle. 3. Find two integers whose sum is 26 and whose product is 165. 4. Two integers have a sum of 16. The sum of their squares is 146. Find the two integers. 5. I am creating a rectangular garden and I have 40 feet of fence to surround it. What is the maximum

area I can get have for my garden? What dimensions will get this area? (Hint, use w and 20-w to represent dimensions of garden)

Alg 3 15


What is the main difference with the fifth problems on the previous page? STEPS: 1. Set up one equation with one unknown -Some of you will want to set up two equations first and use substitution -On these problems, you should not be using linear systems of equations

2. Determine whether you need to solve for the variable or find a maximum/minimum. 3. If solving, use any method (factoring, completing square, quadratic formula) to get roots 4. If finding a maximum/minimum, complete the square, and determine the vertex

6. Find two integers whose sum is 12 and whose product is a maximum. 7. A hot dog vendor sells 120 hot dogs per day at $2 each. He believes that for each $0.25 that he adds

to the price, he will sell 4 less hotdogs. If he wants to maximize his dollars of sales (as determined by number of hotdogs * price), what price should he sell them at? Hint: Let x be the number of $0.25 increases to the price. Therefore, price per hotdog would be 2 + .25 x and the number would be 120 – 4x.

8. The sum of the lengths of the two perpendicular sides of a right triangle is 30 centimeters. What are

the lengths if the square of the hypotenuse is a minimum?

Alg 3 16


10

18

x

x

10+2x

18+2x

Homework Assignment #4 – Quadratic Word Problems

1. One integer is 4 greater than another. The sum of the squares of the integers is 106. Find the

integers. 2. A rectangle is 10 feet by 15 feet. How wide of a border should be added around the entire rectangle

in order to double the area? 3. A backyard swimming pool is rectangular in shape, 10

meters wide and 18 meters long. It is surrounded by a walk of uniform. The walk has an area of 52 square meters. How wide is the walk?

4. A boat travels downstream (with the current) for 36 miles and then makes the return trip upstream

(against the current). The trip downstream took ¾ of an hour less than the trip upstream. If the rate of the current is 4 mph, find the rate of the boat in still water and the time for each part of the trip.

5. The sum of the first n consecutive even positive integers is given by the formula S = n(n+1). How

many consecutive even positive integers must be added to get a sum of 380? 6. The altitude of a triangle is 3 centimeters less than the base to which it is drawn. The area of the

triangle is 44 square centimeters. Find the length of the base.

Alg 3 17


Answers

(1) 5 and 9

(2) 2.5 feet

(3) -7 62+ meters

(4) 20 mph, 1.5 hours downstream, 2.25 hours upstream

(5) 19

(6) 11 centimeters

Alg 3 18


Homework Assignment #5 – Quadratic Word Problems

1. The sum of two numbers is 14. Find the two numbers such that their product is a maximum. (Hint:

Find the maximum of y = x(14-x)). 2. A manufacturer is in the business of producing small models of the Statue of Liberty. He finds that

the daily cost in dollars, C, of producing n statues is given by the quadratic formula

�=��

� 120� 4200. How many statues should be produced per day so that cost will be a minimum? What is the minimal daily cost?

3. A rancher has 600 linear feet of fencing and wants to enclose a field

and then divide it into two equal pastures. What is the maximum area of field that he is able to enclose? Hint: Use the perimeter restriction to substitute into the area formula and get the area formula with only one variable.

4. A person standing near the edge of a cliff 160 feet above the sea throws a rock upwards with an

initial speed of 32 feet per second. The height of the rock above the sea is represented by the

following formula, where t is the time in seconds: � = �16��

32� 160.

a. How many seconds will it take for the rock to reach its maximum height? What is that height? b. At what time will the rock hit the water?

Alg 3 19


Answers

(1) 7 and 7

(2) 60, $600

(3) 15,000 square feet

(4) a. 1 second, reaches 176 feet

b. 1 11+ seconds

Alg 3 20


Quadratic Review Worksheet

(1) Solve for x using any method.

(a) 29x 9x 10 0+ − = (e) 2x 8x 21 0− + =

(b) 22x 6x 3 0− + = (f) 23x 2x 6 0− + =

(c) 2x 6x 247 0+ − = (g) 2x 4x 8 0+ − =

(d) ( )2

3x 1 2x 4+ = + (h) 2x 1 2x 3

= 3x 4 x 3

− +

+ +

(2) Sketch a graph of each of the following. Label the vertex, the axis of symmetry, and at least two other

points.

(a) 8 6x x y 2 +−= (c) 1 6x x3 y 2 +−−=

(b) 1 2x y 2 += (d) 2 x 4x 5y = − + −

(3) A bus service is currently transporting 100 passengers at a cost of $2.00 per person. The owner

estimates that for every 20¢ increase in price there will be 5 fewer passengers. What price should be charged to maximize revenue?

(4) Two integers have a sum of 13. The sum of their squares is 125. What are the numbers? (5) The difference of 2 numbers is 14. Find the numbers if their product is to be a minimum and also find

this product. (6) A rectangle has a length that is 5 less than 3 times the width. Find the dimensions of the rectangle if

the area is 28 cm².

Alg 3 21


Review Answers

(1) (a) 2 5 ,

3 3− (e) 4 i 5±

(b) 3 3

2 2± (f)

1 i 17

3 3±

(c) 13 , −19 (g) 2 2 3− ±

(d) 2 31

9 9− ± (h)

3 i 6

2 2− ±

(3) $3.00 (5 increases of 20 cents) (4) 11 and 2 (5) -7 and 7, Product -49 (6) 4 by 7

Alg 3 22


Additional Review

Change each equation to parabolic form and graph the function.

1. f(x) = x2 - 10x + 25 2. f(x) = x2 + 3

3. f(x) = x2 - 10x + 21 4. f(x) = x2 + 6x + 4

Alg 3 23


5. f(x) = x2 + 4x + 6 6. f(x) = x2 - 4x - 2

GRAPH

7. f(x) = x2 - 4

Vertex: A.O.S.: 8. g(x) = x2 - 4x + 7 Vertex: A.O.S.: 9. m(x) = x2 - 4x + 1 Vertex: A.O.S.:

Alg 3 24


Find the vertex, axis of symmetry, and “a” for each function. Graph each on the grid below.

10. p(x) = x2 + 4x + 1 11. t(x) = -x2 - 4x - 11

V: V: A.O.S. A.O.S. a: a:

12. f(x) = -2x2 - 8x - 11 13. s(x) =2x2 + 8x + 5

V: V: A.O.S. A.O.S. a: a:

14. g(x) =1

4x2+ x − 2

V: A.O.S. a:

Documents

faculty.pingry.orgfaculty.pingry.org/bpoprik/documents/Packet1Quadratics.pdf · Alg 3 4 Quadratic Equations and Parabolas The Quadratic Formula Standard Form: y ax bx c= + +2 2 4