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

Increasing and Decreasing Functions

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.

−∞ ∞ (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.

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.

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?

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.

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.

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.

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

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.

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.

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

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

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.

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