1. Functions Regents Review #3 g(x) = |x – 5| y = ¾x f(x) = 2x – 5
y = -2x2 – 3x + 10
y = (x – 1)2

2. Which relation represents a Function?
Functions
What is a function?
A relation in which every x-value(input) is assigned to exactly one y-value (output)
No x-value is repeated!
Which relation represents a Function? x y 2 6 3 7 4 x y 2 6 7 4
Function
Not a Function

3. Which graph represents a function?
Functions
We 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

4. 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

5. Functions In this course, we explored four different Function Families
Linear Functions
Quadratic Functions
Exponential Functions
Absolute Value Functions

6. Linear Functions Linear Functions “y = mx +b”
The best ways to graph a linear function are…
Table of Values
Slope-Intercept Method

7. Linear Functions Table of Values Method Graph 2x – 4y = 12 y = ½ x – 3-4 -5 -2 -3 2 4 -1
2x – 4y = 12

8. Linear Functions Slope is a ratio: Slope Formula =
Before we discuss the Slope-Intercept Method, let’s discuss SLOPE
Slope is a ratio:
Slope Formula =
0 slope
Undefined slope
Positive Slope
Parallel Lines have the same slope
Negative Slope

9. Linear Functions Slope-Intercept Method y = mx + b m = slope
b = y –intercept (0,b)
Graph 6x + 3y = 9
y = -2x + 3
m =
b = 3 (0, 3)
6x + 3y = 9

10. Linear Functions Horizontal Lines Vertical Lines
y = b where b represents the y-intercept
y = 4
Vertical Lines
x = a where a represents the x-intercept
x = 4
y = 4
x = 4

11. Linear Functions
Writing 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)
Find the y-intercept (b)
y = mx + b Pt.(-3,1)
1 = (-2/3)(-3) + b
1 = 2 + b
-1 = b
Write the equation in “y = mx + b”
y = x – 1
b = -1
m = -2/3

12. Linear Functions
Write 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 = y = 2x + 4
m = 2
Find the y-intercept
y = mx + b Pt.(-2,4)
4 = 2(-2) + b
4 = -4 + b
8 = b
b = 8
Write the equation in “y = mx + b”
y = 2x + 8

13. Quadratic Functions Quadratic Functions “y = ax2 + bx + c”
How do we graph quadratic functions?
Find the coordinates of the vertex
Create a table of values
Graph a parabola
Label the graph with the quadratic equation

14. Quadratic Functions Graph y = x2 + 4x – 12 Vertex = (-2, -16)
Finding the coordinates of the vertex and the axis of symmetry
Finding the x-coordinate
y = x2 + 4x – 12 a = 1, b = 4, c = -12
x =
x = x = -2
Finding the y-coordinate
y = x2 + 4x – 12
y = (-2)2 + 4(-2) – 12
y = 4 – 8 – 12
y = -16
Vertex = (-2, -16)
Also the equation for the axis of symmetry

15. Quadratic Functions Vertex (-2,-16) x-intercept (-6,0) Root: -6
(2,0)
Root: 2 x y 1 -7
-12 -1
-15 -2
-16 -3 -4 -5
f(x) = x2 + 4x – 12
Axis of Symmetry
x = -2

16. Quadratic Functions
The x-values of 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.

17. Quadratic Functions Finding “roots” algebraically
Let’s look at our previous example y = x2 + 4x – 12
In order to find the “roots” , set y equal to zero
y = x2 + 4x – 12
0 = x2 + 4x – 12
0 = (x + 6)(x – 2)
0 = x = x – 2
-6 = x = x
The roots of the function are -6 and 2

18. Quadratic Functions How does a affect the graph of y = ax2 + bx + c ?
If the coefficient of x2 gets larger, the parabola becomes narrower
If the coefficient of x2 gets smaller, the parabola becomes wider
If the coefficient of x2 is negative, the parabola opens downward

19. It represents the y-intercept
Quadratic Functions
How does c affect the graph of y = ax2 + bx + c ? y = x2 y = x2 + 5 y = x2 – 5
It represents the y-intercept

20. Exponential Functions
There are two types of Exponential Functions
Exponential Growth y = abx where b > 1
Exponential Decay y = abx where 0 < b < 1

21. Exponential Functions
Plots:
                                                                                                   
Exponential Functions
y = 2x
y = ½ x x y -2 4 -1 2 1 x y -2 -1 1 2 4

22. Exponential Functions
Properties of Exponential Functions
What happens to y = 2x when….
5 is added multiplied by -1
y = 2x ) y = -2x

23. Exponential Functions
Exponential Growth Formula A = p(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?
a = p(1 + r)t
a = 400( )7
a = 400(1.08)7
a = …
The cost will be $686.00

24. Exponential Functions
Exponential Decay Formula A = p(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?
a = p(1 – r)t
a = 12,900(1 – .14)4
a = 12,900(.86)4
a = …
The cost of the car was $

25. Absolute Value Functions
Absolute Value Functions “y = |x|”
How do you an input an absolute value function into a graphing calculator?
Y =
Math arrow over to NUM
#1 abs(
Input x
Graph

26. Absolute Value Functions
Properties of Absolute Value Functions
What happens to y = |x| when….
5 is added multiplied by -1
y = |x| ) y = -|x|

27. Absolute Value Functions
Properties of Absolute Value Functions
What happens to y = |x| when a number other than 1 is multiplied by x?
As the coefficient of x gets larger, the graph becomes thinner
As the coefficient of x gets smaller, the graph becomes wider

28. Now it’s your turn to review on your own
Now it’s your turn to review on your own! Using the information presented today and your review packet, complete the practice problems in the packet. Regents Review #4 is WEDNESDAY, May 22nd BE THERE!!!!