April 20thReview QuadraticsRational Functions

Exponential Functions

Take out Homework

Page 30 1ad, 3, 5, 12

Families of Functions• Constant• Linear• Quadratic • Rational • Exponential• Step• Periodic• Piecewise

Quadratic Functions• Shows a second degree variation (squared)• Table Graph

• Rule

-2 12

-1 3

0 0

1 3

2 12

𝑦=3 𝑥2 There will always be an exponent of 2

Standard Rule of 2nd Degree

See Page 28 of Textbook:

Finding the Rule1. Use a point on the line (not vertex)

2. Substitute and into rule

3. Solve the resulting equation

4. Write the rule for the function

5. Check

Changing parameter “a” means that the graph would be vertically stretched or vertically compressed.

A closer look at parameter a

The Rule : y = ax²

Exam

pl

es: a = 4

a = 1

a = 1/4

a = -1

What if there’s more to the equation?

http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php

𝑓 (𝑥 )=𝐴𝑥2+𝐵𝑥+𝐶

Rational Functions This function is discontinuous

(not connected) and happens when you have a fraction in the function. “x” has to be the denominator (on the bottom).

Example: A family wanted to get together to buy their mother a new washer and dryer at a cost of $1200. They figured out a rule that would show them how much each family member would pay depending on how many of them participated.

Rational Functions• Is not connected • Table Graph

• Rule

-2 -1.5

-1 -3

0 N/A

1 3

2 1.5

𝑦=3𝑥 There will always be a fraction in the

rule and x will be the denominator

Exponential Functions• The x axis will be an asymptote as • Table Graph

• Rule

-3

-2

-1

0 1

1

2

3 27

𝑦=3𝑥 X will always be the exponent.

Exponential Functions These functions happen

when the x variable is the exponent.

Example: You step on a rusty nail at a construction site, for every 30 seconds you do

not clean out the cut the bacteria in the wound triples.

A closer look at exponential functions

X Y

-3 1/8

-2 1/4

-1 1/2

0 1

1 2

2 4

3 8

TABLE OF VALUES

X2

X2

X2

X2

X2

X2

+1+1+1+1+1+1

For this function a=1

the exponential function

looks like the graph and

has a table of values

like the one below.

Standard Rule of Exponential

Where

Graphically• A curve that passes through (0, a), and approaches

the x-axis on one end while never touching it.

• This line that is approached is called an asymptote.

Parameter “a”• Generates a change in the vertical scale of the

graph

• The further a is from zero, the more the curve is vertically stretched

• The closer a is to zero (decimals), the more the curve is vertically compressed.

• When the sign of a changes, the curve reflects over the x-axis

http://www.analyzemath.com/expfunction/expfunction.html

So what does parameter a do?

• The base of these functions is e=2.81. The blue line a=1The red line a= -1The green line a= -2

What can you concludeabout parameter a?

(there are two things you should notice)

So what does parameter a do?

CONCLUSIONS:

1.If a is positive the graph will increase.

2.If a is negative the graph will decrease.

3.If a is a big number the graph is closer to the y-axis.

4.If a is a small number the graph is closer to the x-axis.

The Base• The value of the base affects its graphical

representation.

• When the base is greater than 1, the curve moves away from the x-axis

• When the base is between 0 and 1, the curve comes closer to the x-axis

http://www.analyzemath.com/expfunction/expfunction.html

A closer look at exponential functions

The Rule: y = a(base)x The base can be any number and the graph will change when the base changes.

Let’s take a look atjust one base...

Finding the Rule1. Substitute the initial value for parameter a.

initial value = y-intercept

2. Sub x and y values of a point on the line (not located on y-axis)

3. Solve equation to determine value of the base

4. Write the rule

5. Check (Validate Solution)

Finding the Rule (Cont)The information you need to be able to find the rule

The initial value (y-intercept)A point on the graph (x, y)The general form of the exponential function

y = a(base)x Initial value = a, therefore:y = 3(base) x Now plug in y=12, x=212 = 3(base) 2 Solve...but how?

Divide both sides by 3, take the square root then you have our base!!!

Examp

le

Question 2

• Turn to pg 42.

http://www.analyzemath.com/expfunction/expfunction.html

Try the concept• Page 42#1aceg, 3, 4, 15