**Unit 5: Nonlinear Functions & Equations**

Write an equation for each parabola in the picture to the right.

Then plug the equation in to your calculator and check to see if the graph matches the one to the right.

What type of graph will each equation below make?

a. y=a(x-m)(x-n) b.y=ax +bx+c

Finding X-Intercepts: Make a graph for the parabola for the parabola given by the equation y= 2x +3x-5. Circle the x intercepts, what are the y-values at the x- intercepts?

*To find the x values for the x-intercept we could solve the equation:

which will yield two different x values since a parabola has two x-intercepts.

Find the points of intersection for the two functions below, then draw a graph showing the points of intersection.

a. y=2x^2+10 & y=-x-5

b. y=x-2 & y=48/x

c. y=4x^2+11x & y=-1x-9

EX 21: Find (x,y)

a. 12x^2-x=-20

b. x^2-10 = 3x

c.-2x^2+11x=12

Review...Or is it?

U5L1I2: Designing Parabolas

Equations for Parabolas:

Factored Form: y=a(x-m)(x-n)

Expanded Form: y=ax +bx+c

U5L1I3&4: FOIL & Un-FOIL

U5L1I4: Finding Solutions and Intersections

Ex 22: Use factoring to solve for x in each expression below.

U5L1I3:Reverse FOIL (Factoring Trinomials)

U5L1I4: Finding x-intercepts and Solutions.

Writing Equations for Parabolas

EXAMPLES

To help you remember

Use the quadratic formula to find the x intercepts for each parabola below.

A. y=4x^2+20x+25

B. y=2x^2-3x-14

U5L2I1: The Quadratic Formula

Ex.22

**Unit 5 Intro:**

What is an Intercept?

What is an Intercept?

**Getting Started:**

Circle the x intercept and box y intercept for each graph. Then write down the ordered pair for each.

Circle the x intercept and box y intercept for each graph. Then write down the ordered pair for each.

Four equations are shown below use algebra to find the x and y intercept for each equation.

Ex. 1

A. 3x+2=y B. 3x+5y=30

C. y=-4x+2 D. 4x-5y=20

Ex. 2

Use the x and y intercept to graph each equation.

Match each graph below to I the corresponding equation.

Ex. 3

A.y=-1-x

B.y=2-1x

C.y=2x+1

D.y=-2x+4

E.y=1x+2

F.y=3x-3

A. f(x) = -2x+10

i. f(2)=

ii. f(3)=

iii. f(-1)=

iv. f(x)=8

v. f(x)=-6

Ex 3: Which table could represent a function? Explain

U5L1I1: What is a function?

Ex 5: Function Notation

Ex. 6, Ex.7 & Ex. 8

Ex 4. Which graph could represent a function. Explain

D. Create a Table that does not represent a function.

E. Create a table that does represent a function.

Draw a graph of a function and a graph that would not be a function.

Use each of the given function to do the calculations below.

B. g(x)= 4/x

i. g(1)=

ii. g(3)=

iii. g(-2)=

iv. g(x)=2

v. g(x)=-12

Ex 6: Mr. Branch has designed a “toy” catapult to use for a Physics Demonstration. the catapult launches a tennis ball with the same velocity every time it is fired, from a height of 0.75m. Mr. Branch uses the following equation to determine the height of the ball, h after various times, t.; h(t) = -4.9t^2+23t+0.75.

A. Do you think this equation is a function? Use you calculator to make a graph to justify your answer. (hint: set your y max to 50)

B. Find the height of the tennis ball, h(t) for each time below:

i. t= 1.5s

ii. t= 4s

iii. t=-1s

C. Use your table to figure out when h(t)=0, what is significant about this point for the path of the tennis ball.

Ex7: The leadership class is trying to figure out how much to charge, c, for homecoming tickets. The total cost of the DJ, decorations, snacks and photographer is $950, so they will use the equation c(n) = 950/n, where n is the number of students they think will attend.

A. Do you think this equation is a function? Use you calculator to make a graph to justify your answer. (hint: set your y max to 1000 and y min to -1000)

B. Find the cost of tickets, c(n) for each projected number of students:

i. n= 300

ii. n=100

iii. n=-50

Ex 8: Go back and determine the theoretical and practical domain and range for examples 5-7

What is a function & Vertical line test

Function Notation, Domain & Range

Each graph below contains a solid line and a dashed line. Put a circle on the x-int, a square on the maximum or minimum value, and find the ordered pair for each. Then Identify the domain and range for each graph.

Graph 1

Graph 2

Graph 3

Examine the x values of the x-int and the max or min, do you notice any relationship? Does this work everytime?

y=a(x-m)(x-n)

EX. 9

Write an equation for each parabola below, then use your calculator to check if your equation

Graph 1

Graph 2

Ex. 12

Write an equation for each parabola described, you are given the x value for the xint and the ordered pair for the max or min. Then draw the parabola on the graph provided.

A. X int:1 & -1

Min:(0,-8)

B. X int:-4 & 2

Max:(-1,9)

C. X int:-1 & 7

Max: (3,8)

D. X int:4

y int: (0,4)

Ex. 13

A.The parabola has a y-intercept of 12, write an equation for this parabola.

B.What is the x coordinate of the maximum?

C. What is the y coordinate of the maximum.

A parabola has x intercepts at 8 and -6.

Ex 10. Write an equation for each parabola below.

Ex. 11

Find the equation for the portion of each parabola shown below.

Find the coordinates for the max or min for each parabola.

2

If we have an equation in factored form, how can we re-write it in expanded form?

FOIL

F- First

O- Outside

I- Insides

L- Lasts

When you foil you are finding the product of two binomials.

Ex. 14: Find the product of each set of binomials.

a.(x+5)(x+6)

b.(x-3)(x+9)

c.(x-2)^2

d.(x-10)(x+10)

e.(x+1)(x-5)

f.(X+8)^2

g.(x-2)(x+1)

h.(x-a)(x+a)

i.(x+a)(x+b)

j.(x+a)(x-b)

Factors

Factors are numbers you can multiply together to get another number.

Ex.15: Make a list of all the factors of each number

a. 18

b. 40

c. 36x

d. 39y

e. 90

f. 4xy

The Greatest Common Factor (GCF)

The largest number that is the factor of two numbers.

a. Ex15 a & b

b. Ex15 b & e

Ex.16: Find the GCF for the two numbers.

c.9,27

d.16, 64

e.x , 4x

f.3x , 36

2

2

2

2

What are the benefits of each?

Steps to Factoring Box Method

ax

c

#bx

Factor: x +5x+6

2

GCF

GCF

GCF

GCF

1.Multiply a and c

(1 & 6)

2.List the factors of a*c

3.Find the factors of a*c that add to b. (#bx)

4.Rewrite the original equation to include the factors of a*c.

(ax +#bx+#bx+c)

5.Fill in the four boxes as shown.

6.Find the GCF of each column and row.

7.The left side and top make up the two binomial factors.

Steps

#bx

2

EX 17. Factoring Trinomials with a=1

a.x +9x+18 b. x -4x+3

C.x -2x-15 d. x +4x-5

e.x +11x+18 f. x +x-20

g.x -64 h. x -6x+9

i.x +8x+16 j. x +10x-11

k.x -6x-16 L. x -8x+12

2

2

2

2

2

2

2

2

2

2

2

2

2

EX 18. Factoring Trinomials with a=1

a.2x -x-3 b. 12x -5x-2

C.12x +10x+2 d. 10x +13x-3

e.9x -64 f. 3x -11x+6

g.-2x -7x+6 h. 3x +12x

i.6x +23x+20 j. -3x -13x-12

k.36x -25 L. 4x -24x

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Ex. 19: Find the x-intercepts for each parabola below.

A. y= -x +x+12 B. y= x -3x-10

C. y= -5x +28x-32 D. y= 10x -x-24

E. y= x +6x+9

Ex20. Review: Look back at ex19 a and b, find the y intercept and max or min for each.

EX.19 and 20

2

2

2

2

2