1. More on Functions and Their Graphs
Increasing and Decreasing Functions
Relative Maxima and Relative Minima
Even and Odd Functions and Symmetry
Functions and Difference Quotients

2. Increasing and Decreasing Functions

4. State the intervals on which the given function is increasing, decreasing, and constant.
−5, −2
(0, 2)
(5, ∞)
Decreasing:
−∞, −5
−2, 0
(2,5)
(−𝟐,𝟐)
(𝟐,𝟐) −∞ ∞
(𝟎,𝟎)
Constant:
none
(−𝟓, −𝟓)
(𝟓,−𝟓)
Go from left to right like you’re walking a trail.
Remember to use x-coordinates.

5. −∞ ∞ (𝟐, 𝟒) Go from left to right like you’re walking a trail.
Increasing:
(2, ∞)
Decreasing:
(−∞, 2) −∞ ∞
Constant: none
(𝟐, 𝟒)
Go from left to right like you’re walking a trail.
Remember to use x-coordinates.

6. Relative Maxima and Relative Minima
(Also known as local maxima and local minima)
Keep it simple. These are the points at which the function changes its increasing or decreasing behavior. They are the turning points.

7. Where are the relative maxima? Where are the relative minima?
(−𝟐,𝟐)
(𝟐,𝟐) −∞ ∞
(𝟎,𝟎)
(−𝟓, −𝟓)
(𝟓,−𝟓)
Where are the turning points?

8. Even and Odd Functions and Symmetry
A function is an even function if 𝑓 −𝑥 =𝑓 𝑥 . The graph shows symmetry with respect to the y-axis.
𝒇 𝒙 = 𝒙 𝟐
The coordinates are 𝑥, 𝑦 and (−𝑥,𝑦). The right side of the equation does not change when 𝑥 is replaced with −𝑥.
The left and right sides of the graph are reflections of each other.
The function passes through the origin.
Notice that the y-coordinate stays the same.

9. The alternate opposite sides show symmetry to each other.
A function is odd if 𝑓 −𝑥 =−𝑓 𝑥 . The graph shows symmetry with respect to the origin.
The coordinates are 𝑥, 𝑦 and (−𝑥,− 𝑦). The right side of the equation does not change when 𝑥 is replaced with −𝑥. Every term on the right side of the equation changes its sign if 𝑥 is replaced with −𝑥. Notice the signs of the coordinates.
𝒇 𝒙 = 𝒙 𝟑
The alternate opposite sides show symmetry to each other.
The function passes through the origin.

10. It passes through the origin.
(−𝟐, 𝟐)
(𝟐, 𝟐)
It is an even function.
(−𝟓, −𝟓)
(𝟓, −𝟓)
It passes through the origin.
It is symmetrical on both sides of the y-axis.

11.   (−𝟒, 𝟒) The function is odd. (𝟒, −𝟒) It passes through the origin.
The alternate opposite sides show symmetry to each other.

12. EVEN FUNCTIONS
Algebraically: f(-x) = f(x) for all x in the domain of f. This means you take the function and plug in –x for x. If you end up with the original equation, it is an even function or symmetric with respect to the y-axis.

13. ODD FUNCTIONS Algebraically: f(-x) = -f(x) for all x in the domain of f. This means you take the function and plug in –x for x. If you end up with the opposite of the original equation – it is an odd function or symmetric with respect to the origin.

14. Verifying Algebraically….
Copy the original function
Replace −𝑥 for every 𝑥 variable
Simplify the function
Compare to the original to determine its symmetry
Make a concluding statement with an algebraic statement to support it.
𝑓 −𝑥 =𝑓(𝑥) EVEN
𝑓 −𝑥 =−𝑓(𝑥) ODD
𝑓 −𝑥 ≠𝑓 𝑥 ≠−𝑓(𝑥) NEITHER

15. Are these functions even, odd, or neither? 𝑓 𝑥 = 𝑥 3 −6𝑥
𝑔 𝑥 = 𝑥 4 −2 𝑥 2
ℎ 𝑥 = 𝑥 2 +2𝑥+1
Odd
Even
Neither

16. The Difference Quotient
The difference quotient of a function f is an expression of the form
where h ≠ 0.
Where does it come from?
The difference quotient, allows you to find the slope of any curve or line at any single point.

17. Functions and Difference Quotient
Steps Used in Finding A Difference Quotient
Find 𝑓 𝑥+ℎ , that is, substitute (𝑥+ℎ) into every 𝑥.
Simplify 𝑓 𝑥+ℎ −𝑓 𝑥 . Note: 𝑓(𝑥) is the given equation.
Divide the result by ℎ