Regents Review #3

Functionsf(x

) = 2

x – 5

y = -2x2 – 3x + 10

g(x) = |x – 5

|

y = ¾x

y = (x – 1)2 Roslyn Middle SchoolResearch Honors Integrated Algebra

FunctionsWhat is a function?

A relation in which every x-value(input) is assigned to exactly one y-value (output)

Which relation represents a Function?

x y

2 6

3 7

4 7

x y

2 6

2 7

4 7

Function Not a Function

FunctionsWe can recognize functions using the vertical line test

Vertical Line Test: If a graph intersects a vertical line in more than one place, the graph is not a function

Which graph represents a function?

Function Not a function

Functions

Functions can be written using function notation “f(x)” means f of x

Example: f(x) = 2x – 3 means the same as y = 2x – 3 g(x) = 2x – 3 means the same as y = 2x – 3

h(x) = 2x – 3 means the same as y = 2x – 3

Functions

In this course, we explored four different Function Families

1)Linear Functions

2)Quadratic Functions

3)Exponential Functions

4)Absolute Value Functions

Linear Functions

Linear Functions “y = mx +b”

The best ways to graph a linear function are…

1)Table of Values

2)Slope-Intercept Method (most effective)

Linear FunctionsTable of Values MethodGraph 2x – 4y = 12 y = ½ x – 3

x y

-4 -5

-2 -4

0 -3

2 -2

4 -1

2x – 4y = 12

Linear FunctionsBefore we discuss the Slope-Intercept Method, let’s discuss SLOPE

Slope is a ratio:

Slope Formula =

Run

Rise

12

12

xx

yy

Positive Slope

Negative Slope

0 slope

Undefined slope

Parallel Lines have the same slope

Perpendicular lines have opposite, reciprocal slopes

Linear FunctionsSlope-intercept Method y = mx + b m = slope b = y –intercept (0,b)

Graph 6x + 3y = 9

y = -2x + 3

m =

b = 3 (0, 3)

1

2

6x + 3y = 9

Linear FunctionsHorizontal Linesy = b where b represents the y-intercept

y = 4

Vertical Linesx = a where a represents the x-intercept

x = 4

y = 4 x = 4

Linear FunctionsWriting the Equation of a Line

Write the equation of a line that runs through the points (-3,1) and (0,-1)

Find the slope (m)

(-3,1) (0,-1)

12

12

xx

yym

3

2

)3(0

11

m

Find the y-intercept (b)

y = mx + b

Write the equation in “y = mx + b”

y = x – 1 3

2

m = -2/3

1 = (-2/3)(-3) + b

1 = 2 + b

-1 = b

b = -1

Linear FunctionsWrite the equation of a line that is parallel to y – 2x = 4 and runs through the point (-2,4)

Find the slope

Parallel lines have the same slope

y – 2x = 4 y = 2x + 4

m = 2

Find the y-intercept

y = mx + b

4 = 2(-2) + b

4 = -4 + b

8 = b

b = 8

Write the equation in “y = mx + b”

y = 2x + 8

Quadratic Functions

Quadratic Functions “y = ax2 + bx + c”

How do we graph quadratic functions?

1)Find the coordinates of the vertex 2)Create a table of values3)Graph a parabola4)Label the graph with the quadratic equation

Quadratic FunctionsGraph f(x) = x2 + 4x – 12

f(x) = x2 + 4x – 12 means y = x2 + 4x – 12

Finding the coordinates of the vertex

Finding the x-coordinate

y = x2 + 4x – 12 a = 1, b = 4, c = -12

x =

x = x = -2

a

b

2

22

4

)1(2

4

Finding the y-coordinate

y = x2 + 4x – 12

y = (-2)2 + 4(-2) – 12

y = 4 – 8 – 12

y = -16

Vertex = (-2, -16)

Quadratic Functions

Vertex (-2,-16)

x y

1 -7

0 -12

-1 -15

-2 -16

-3 -15

-4 -12

-5 -7

x-intercept(-6,0)

x-intercept(2,0)

Axis of Symmetryx = -2

f(x) = x2 + 4x – 12

Quadratic Functions

The x-intercepts of the graph of a quadratic function are also known as the “roots”.

We can identify the “roots” of a quadratic function by looking at the graph of a parabola and locating the x-intercepts.

We can also identify the roots algebraically.

Quadratic Functions

Finding “roots” algebraically

Let’s look at our previous example y = x2 + 4x – 12

In order to find the “roots” (x-intercepts), set y = to zero

y = x2 + 4x – 12 0 = x2 + 4x – 12 0 = (x + 6)(x – 2)0 = x + 6 0 = x – 2 -6 = x 2 = x

The roots of the function are (0,-6) and (0,2)

Quadratic FunctionsHow does a affect the graph of y = ax2 + bx + c ?

1. If the coefficient of x2 gets larger, the parabola becomes narrower

2. If the coefficient of x2 gets smaller, the parabola becomes wider

3. If the coefficient of x2 is negative, the parabola opens downward

Quadratic Functions

How does c affect the graph of y = ax2 + bx + c ?

y = x2 y = x2 + 5 y = x2 – 5

“c” represents the y-intercept and moves the parabola up and down.

Exponential Functions

There are two types of Exponential Functions

1)Exponential Growth y = abx where b > 1

2)Exponential Decay y = abx where 0 < b < 1

Exponential Functions

y = 2x y = ½ x

Plots:

                                                                                                   

x y

-2 ¼

-1 ½

0 1

1 2

2 4

x y

-2 4

-1 2

0 1

1 ½

2 ¼

Exponential Functions

Properties of Exponential FunctionsWhat happens to y = 2x when…. 5 is added multiplied by -11)y = 2x + 5 2) y = -2x

Exponential FunctionsExponential Growth Formula y = a(1 + r)t

The cost of maintenance on an automobile increases each year by 8%. If Alberto paid $400 this year for maintenance for his car, what will the cost be (to the nearest dollar) seven years from now?

y = a(1 + r)t

y = 400(1 + .08)7

y = 400(1.08)7

y = 685.5297…

The cost will be $686.00

Exponential FunctionsExponential Decay Formula y = a(1 – r)t

A used car was purchased in July 1999 for $12,900. If the car loses 14% of its value each year, what was the value of the car (to the nearest penny) in July 2003?

y = a(1 – r)t

y = 12,900(1 – .14)4

y = 12,900(.86)4

y = 7056.4052…

The cost of the car was $7056.41

Absolute Value Functions

Absolute Value Functions “y = |x|”How do you an input an absolute value function into a graphing calculator?

1)Y =

2)Math arrow over to NUM

3)#1 abs(

4)Input x

5)Graph

Absolute Value Functions

Properties of Absolute Value FunctionsWhat happens to y = |x| when…. 5 is added multiplied by -1 1)y = |x| + 5 2) y = -|x|

Absolute Value FunctionsProperties of Absolute Value Functions

What happens to y = |x| when a number other than 1 is multiplied by x?

1. As the coefficient of x gets larger, the graph becomes thinner

2. As the coefficient of x gets smaller, the graph becomes wider

Regents Review #3

Now it’s time to study!

Using the information from this power point and your review packet,

complete the practice problems.