1. Welcome! Section 3: Introduction to Functions Topic 6, 8-11 Topics 6
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Homework: Section 3 Study Guide
Study for Exam Thursday B Day 11/17
Topics 8-11

2. Section 3: Introduction to Functions Topic 6 - 8
Students will:
Explore the closure property with polynomials and discover polynomials are closed under the same operations as integers.
Explore solutions and 𝑦 intercepts of functions, when functions are increasing and decreasing, and relative maximums and minimums.

3. The Counting Numbers {1, 2, 3, ...}
Section 3: Introduction to Functions Topic 1 - 2
Section 3: Introduction to Functions Topic 6 - 8
Let’s Review the world of numbers and general order of discovery.
The simplest idea of a number is something to count with.
The Counting Numbers {1, 2, 3, ...}

4. Section 3: Introduction to Functions Topic 1 - 2
Zero
The idea of zero, though natural to us now, was not natural to early humans ... if there is nothing to count, how can you count it?
Whole Numbers {0, 1, 2, 3, ...}

5. Section 3: Introduction to Functions Topic 1 - 2
Negative Numbers
We can count forward and we can count backwards! How? Temperatures can be less than zero, we can owe money…so negative numbers exist.
Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}

6. {p/q : p and q are integers, q is not zero}
Section 3: Introduction to Functions Topic 1 - 2
Section 3: Introduction to Functions Topic 6 - 8
Fractions  Ratios
If you have an apple and you want to share it…
You took 1 and divided it by….
Rational Numbers:
{p/q : p and q are integers, q is not zero}

7. Numbers that can’t be fractions
Section 3: Introduction to Functions Topic 1 - 2
Section 3: Introduction to Functions Topic 6 - 8
Numbers that can’t be fractions
Like the square root of 2 (√2) and Pi (π). They are useful to find diagonal distances across squares; great in calculations of circles, π
So we need and have
Irrational Numbers

8. Real Numbers Section 3: Introduction to Functions Topic 1 - 2
Real Numbers include:
the rational numbers, and
the irrational numbers

9. Closure Property: Section 3: Introduction to Functions Topic 1 - 2

10. Closure Property: Section 3: Introduction to Functions Topic 1 - 2
Sets
A set is a collection of things (usually numbers).
Examples:
Set of even numbers: {..., -4,-2, 0, 2, 4, ...}
Set of odd numbers: {..., -3, -1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

11. Is the set of odd numbers closed under
Section 3: Introduction to Functions Topic 1 - 2
Section 3: Introduction to Functions Topic 6 - 8
Let’s look at a set of odd numbers:
Example: Odd numbers {..., -3, -1, 1, 3, ...}
Is the set of odd numbers closed under
the operations of + − × ÷ ?

12. Odd Numbers: Section 3: Introduction to Functions Topic 1 - 2
Adding?
3 + 7 =
Subtracting?
11 − 3 =
Multiplying?
5 × 7 =
Dividing?
33/3 = /5 =

13. Worksheet Practice Section 3: Introduction to Functions Topic 1 - 2

14. Section 3: Introduction to Functions Topic 6 - 8
The y-intercept is the y-coordinate of the point where the graph intersects the y-axis. The x-coordinate of this point is always 0.
The x-intercept is the x-coordinate of the point where the graph intersects the x-axis. The y-coordinate of this point is always 0.

15. Example 1A: Finding Intercepts
Section 3: Introduction to Functions Topic 6 - 8
Example 1A: Finding Intercepts
Find the x- and y-intercepts.
The graph intersects the y-axis at (0, 1).
The y-intercept is 1.
The graph intersects the x-axis at (–2, 0).
The x-intercept is –2.

16. Section 3: Introduction to Functions Topic 6 - 8
Check It Out! Example 1a
Find the x- and y-intercepts.
The graph intersects the y-axis at (0, 3).
The y-intercept is 3.
The graph intersects the x-axis at (–2, 0).
The x-intercept is –2.

17. Example 1B: Finding Intercepts
Section 3: Introduction to Functions Topic 6 - 8
Example 1B: Finding Intercepts
Find the x- and y-intercepts.
5x – 2y = 10
To find the x-intercept, replace y with 0 and solve for x.
To find the y-intercept, replace x with 0 and solve for y.
5x – 2y = 10
5x – 2(0) = 10
5x – 2y = 10
5(0) – 2y = 10
5x – 0 = 10
5x = 10
0 – 2y = 10
– 2y = 10
x = 2
The x-intercept is 2.
y = –5
The y-intercept is –5.

18. Section 3: Introduction to Functions Topic 6 - 8
Check It Out! Example 1a
Find the x- and y-intercepts.
The graph intersects the y-axis at (0, 3).
The y-intercept is 3.
The graph intersects the x-axis at (–2, 0).
The x-intercept is –2.

19. Section 3: Introduction to Functions Topic 6 - 8
Check It Out! Example 1b
Find the x- and y-intercepts.
–3x + 5y = 30
To find the x-intercept, replace y with 0 and solve for x.
To find the y-intercept, replace x with 0 and solve for y.
–3x + 5y = 30
–3x + 5(0) = 30
–3x + 5y = 30
–3(0) + 5y = 30
–3x – 0 = 30
–3x = 30
0 + 5y = 30
5y = 30
x = –10
The x-intercept is –10.
y = 6
The y-intercept is 6.

20. Section 3: Introduction to Functions Topic 6 - 8
Check It Out! Example 1c
Find the x- and y-intercepts.
4x + 2y = 16
To find the x-intercept, replace y with 0 and solve for x.
To find the y-intercept, replace x with 0 and solve for y.
4x + 2y = 16
4x + 2(0) = 16
4x + 2y = 16
4(0) + 2y = 16
4x + 0 = 16
4x = 16
0 + 2y = 16
2y = 16
x = 4
The x-intercept is 4.
y = 8
The y-intercept is 8.

21. Section 3: Introduction to Functions Topic 6 - 8
Remember, to graph a linear function, you need to plot only two ordered pairs. It is often simplest to find the ordered pairs that contain the intercepts.
Helpful Hint
You can use a third point to check your line. Either choose a point from your graph and check it in the equation, or use the equation to generate a point and check that it is on your graph.

22. Example 3A: Graphing Linear Equations by Using Intercepts
Section 3: Introduction to Functions Topic 6 - 8
Example 3A: Graphing Linear Equations by Using Intercepts
Use intercepts to graph the line described by the equation.
3x – 7y = 21
Step 1 Find the intercepts.
y-intercept:
x-intercept:
3x – 7y = 21
3x – 7(0) = 21
3x – 7y = 21
3(0) – 7y = 21
3x = 21
–7y = 21
x = 7
y = –3

23. Section 3: Introduction to Functions Topic 6 - 8
Example 3A Continued
Use intercepts to graph the line described by the equation.
3x – 7y = 21
Step 2 Graph the line.
Plot (7, 0) and (0, –3). x
Connect with a straight line.

24. Section 3: Introduction to Functions Topic 9-11
Common Functions

25. Section 3: Introduction to Functions Topic 9-11
Increasing Functions
A function is "increasing" when the y value 
increases as the x-value increases, like this:

26. Section 3: Introduction to Functions Topic 9-11
No Slope  Zero slope

27. Section 3: Introduction to Functions Topic 9-11
Undefined

28. Section 3: Introduction to Functions Topic 9-11
Local Maximum and Minimum
Functions can have “hills and valleys": places where they reach a minimum or maximum value.
It may not be the minimum or maximum for the whole function, but relatively it is.

29. Function Transformations
Section 3: Introduction to Functions Topic 9-11
Function Transformations
Let us start with a function, in this case it is f(x) = x2, but it could be anything:
add to y, go high
F(x)

30. Function Transformations
Section 3: Introduction to Functions Topic 9-11
Function Transformations
Let us start with a function, in this case it is f(x) = x2, but it could be anything:
add to x, go left
F(x)
F(x)

31. Section 3: Introduction to Functions Topic 9-11
Function Transformations
An easy way to remember what happens to the graph when we add a constant:
add to y, go high add to x, go left
f(x) + C
C > 0 moves it up
C < 0 moves it down
f(x + C)
C > 0 moves it left
C < 0 moves it right

32. Section 3: Introduction to Functions Topic 9-11
Piecewise Functions
A Function Can be in Pieces
We can create functions that behave differently based on the input (x) value.
A function made up of
3 pieces

33. Section 3: Introduction to Functions Topic 9-11
Example:
when x is less than 2, it gives x2,
when x is exactly 2 it gives 6
when x is more than 2 and less than or equal to 6 it gives the line 10-x
It looks like this:
(a solid dot means "including", an open dot means "not including")

34. Section 3: Introduction to Functions Topic 9-11
This is how we write it: X
Function Y -4 16 -2 4 1 2 6 3 7

35. Section 3: Introduction to Functions Topic 1 - 2
Homework
Section 3 Study Guide