1. Mod. 3 Day 4
Graphs of Functions

2. Relative Extremes on a Graph
Relative Maximum: A “peak” point on a graph. The highest point in a particular section of a graph.
Relative Minimum: A “valley” point on a graph. The lowest point in a particular section of a graph.

3. Examples of graphs Rel. Max Rel. Max Rel. Min Rel. Min Rel. Min

4. Multiplicity
When a factor is repeated, it touches the graph multiple times at the same root.
Example: x² is the same as (x-0)², so the function would have two roots where x = 0

5. Simplifying Radicals Examples: 72 80 45 200 75 216 6 2 4 5 3 5 10 2
6 2 80
4 5 45
3 5
200
10 2 75
5 3
216
6 6

6. Quadratic Formula x= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
Identify 𝑎,𝑏,𝑐 and plug them in the formula
Simplify and solve
You will end up with 2 solutions.
If you get the same solution twice, that is multiplicity.
If you have a negative value under the radical, that is an imaginary number
Examples:
2𝑥 2 +5𝑥−12
𝑥=−4, 3 2
𝑥 2 +4
𝑥=±2𝑖

7. Imaginary Numbers 𝑖= −1 𝑖 2 =−1 Example: 5𝑖∙2𝑖 Solution: −10
𝑖= −1
𝑖 2 =−1
Example: 5𝑖∙2𝑖
Solution: −10
Example: −3𝑖(4+5𝑖)
Solution: −12𝑖+15

8. (𝑎 + 𝑏𝑖) and (𝑎 – 𝑏𝑖) are conjugate pairs.
Conjugates
(𝑎 + 𝑏𝑖) and (𝑎 – 𝑏𝑖) are conjugate pairs.
Example: (4+2𝑖) and (4−2𝑖) are conjugates
Multiply/FOIL: 16 −8𝑖 +8𝑖 −4 𝑖 2
16+4 = 20

9. Multiply Complex Numbers
Given: 2+𝑖 , create the missing conjugate and multiply
(2+𝑖)(2−𝑖) =
4 − 𝑖²
4 + 1 5

10. Where do conjugates come from?
Find the solutions to the equation: x² - 4x + 15=0
Can’t factor to solve, therefore use quadratic formula
Use quadratic equation: 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
Solution: 2 ± 𝑖 11
So the solutions are conjugates!
2 + 𝑖 11 and 2 − 𝑖 11

11. Real vs. Imaginary Solutions:

12. Real vs. Imaginary Solutions:
How many turns? 4
Degree?
Solutions
2 real solutions
2 imaginary solutions

13. Creating Factors and Foiling
Given the following zeros, create the polynomial:
1, -5, 6
Step 1: Create the factors
(x-1)(x+5)(x-6)
Step 2: Foil the first two
( 𝑥 2 +4𝑥−5)
Step 3: Multiply by the last factor
𝑥 2 +4𝑥−5 x−6
𝑥 3 −2 𝑥 2 −29𝑥+30

14. Creating Factors and Foiling
Given the following zeros, create the polynomial:
2, -3i
Step 1: Create the missing conjugate 3i
Step 2: Create the factors
(x-2)(x+3i)(x-3i)
Step 2: Foil the conjugates
𝑥+3𝑖 𝑥−3𝑖
𝑥 2 +9
Step 3: Multiply by the last factor
𝑥 x−2
𝑥 3 −2 𝑥 2 +9𝑥−18

15. Creating Factors and Foiling
Given the following zeros, create the polynomial:
7, 5i, -5i
Step 1: Create the factors
(x-7)(x-5i)(x+5i)
Step 2: Foil the conjugates
𝑥−5𝑖 𝑥+5𝑖
𝑥 2 +25
Step 3: Multiply by the last factor
𝑥 x−7
𝑥 3 −7 𝑥 2 +25𝑥−175

16. Clickers

17. Multiplying Conjugates To Create A Polynomial Equation:
Example: (x + 3i)(x-3i)
Foil the conjugates
𝑥 2 −3𝑥𝑖+3𝑥𝑖−9 𝑖 2
𝑥 2 −9 𝑖 2
Polynomial Equation: 𝑥 2 +9
Example: (3x+4i)(3x-4i)
9𝑥 2 −12𝑥𝑖+12𝑥𝑖−16 𝑖 2
9𝑥 2 −16 𝑖 2
Polynomial Equation : 9𝑥2+16

18. Even Degree End Behavior of a Graph Odd Degree Positive Coefficient:
as x → -∞ f(x) → -∞
as x → ∞ f(x) → ∞
Negative Coefficient:
as x → ∞ f(x) → -∞
Positive Coefficient:
as x → -∞ f(x) → -∞
as x → ∞ f(x) → ∞
Negative Coefficient:
as x → -∞ f(x) → ∞
as x → ∞ f(x) → -∞

19. Roots, Solutions, or Zeros of Functions
A function has as many complex solutions equal to the degree of the function.
A real root, solution or zero of a graph can be found where the function crosses the x-axis.
An imaginary function can not be seen on a graph.

20. Multiply Complex Numbers
(5+ 2𝑖 )(5− 2𝑖 )=
Solution: 25 − 4 𝑖 2 = 25 – 2𝑖
(3+7 2𝑖 )( 3−7 2𝑖 )=
Solution: 9 − 𝑖 2 = 9 – 98𝑖