Section 1.3 More on Functions and Their Graphs

The open intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates.

Increasing, Decreasing, Constant is just like riding a roller coaster! Example 1 Find where the graph is increasing? Where is it decreasing? Where is it constant? Increasing: (-5, -2) Decreasing: (-∞, -5) Constant: (0, 2) No where (-2, 0) (5, ∞) (2, 5) Increasing, Decreasing, Constant is just like riding a roller coaster! Increasing Decreasing Increasing Decreasing Increasing Decreasing

Example 2 Find where the graph is increasing? Where is it decreasing? Where is it constant? (2, ∞) (-∞, 2) Constant: Decreasing: Increasing: No where Increasing Decreasing

Example 3 Find where the graph is increasing? Where is it decreasing? Where is it constant? Increasing: Decreasing: Constant: [-2, 0) [0, 2) [2, 4) No Where No Where Constant Constant Constant

Relative Maxima And Relative Minima

Example 4 Where are the relative minimums? Where are the relative maximums? Why are the maximums and minimums called relative or local? (-2, 2) (2, 2) (-5, -5) (5, -5)

Even and Odd Functions and Symmetry

Even and Odd Functions and Symmetry A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (- x, y) is also on the graph. All even functions have graphs with this kind of symmetry. A graph is symmetric with respect to the origin if for every point (x, y) on the graph, the point (-x, -y) is also on the graph. All odd functions have graphs with origin symmetry (or 180 degree rotational symmetry).

It matches up with mirror symmetry over the y-axis. Example 5 Is this an even or odd function? Even! It matches up with mirror symmetry over the y-axis.

It has 𝟏𝟖𝟎 𝟎 rotational symmetry. Example 6 Is this an even or odd function? Odd! It has 𝟏𝟖𝟎 𝟎 rotational symmetry.

It has 𝟏𝟖𝟎 𝟎 rotational symmetry. Example Is this an even or odd function? Odd! It has 𝟏𝟖𝟎 𝟎 rotational symmetry.

Even or Odd Algebraically Even functions are when Odd functions are when Examples 1. 2. Not the Same, so not EVEN. Now factor out a -1. The Same, so it’s EVEN. Now it’s the Same, so it’s ODD.

Even and Odd Algebraically Neither. Neither. Odd. Neither. Odd.

Interpreting Information based on a graph For the graph pictured, find the following The domain of f The range of f The x-intercepts The y-intercepts Intervals where f is increasing Intervals where f is decreasing Intervals where f is constant The point at which f has a relative minimum/maximum (-∞, ∞) (-2, 2) and (2, 2) [-5, ∞) maximums (-6.2, 0), (-3, 0), (0, 0), (3, 0), (6.2, 0) (0, 0) (-5, -2), (0, 2), (5, ∞) minimums (-5, -5) and (5, -5) (-∞, -5), (-2, 0), (2, 5) No where. f(-2) = 2 and f(2) = 2 and f(6.5) = 2 and f(-6.5) = 2 The values for f (-3) = ? and f (x) = 2 Is f even, odd, or neither? f(-3) = 0 Even.

Piecewise Functions A function that is defined by two or more equations over a specific domain is called piecewise function. Many cell phone plans can be represented with piecewise functions. Example Find and interpret each of the following.

Example Graph the following piecewise function.

Piecewise Functions – Step Function – Greatest Integer Function Some piecewise functions are called STEP FUNCTIONS because their graph form DISCONTINUOUS steps. Check out the picture! One such function is called the GREATEST INTEGER FUNCTION, which is symbolized by int(x) or [x]. This means the greatest integer that is less than or equal to x. 1 1 1 1 2 2 2 2

More examples! Please add them in to your paper. = 3 = 94 = -7 = 29

Weight For Letters (In ounces) Example The charge would be $0.59. The charge would be $0.76. $1.25 $1.00 $0.75 Pricing For Letters (In US dollars) $0.50 $0.25 1 2 3 4 5 Weight For Letters (In ounces)

Functions and Difference Quotients See next slide.

Continued on the next slide.

Example Find and simplify the expressions if

Example Find and simplify the expressions if

Example Find and simplify the expressions if

Time to review (a) (b) (c) (d) (c)

(a) (b) (c) (d) (a)

(a) (b) (c) (d) (c)