3 2 Graphs of Functions OBJECTIVES 1 DETERMINE

3. 2 Graphs of Functions OBJECTIVES: 1. DETERMINE WHETHER A GRAPH REPRESENTS A FUNCTION. 2. ANALYZE GRAPHS TO DETERMINE DOMAIN AND RANGE, LOCAL MAXIMA AND MINIMA, INFLECTION POINTS, AND INTERVALS WHERE THEY ARE INCREASING, DECREASING, CONCAVE UP, AND CONCAVE DOWN. 3. GRAPH PARAMETRIC EQUATIONS.

Example #1 A Function Defined by a Graph The graph defines the function f. Determine the following: A. f(0) − 2 B. f(2) 1 C. The domain of f. D. The range of f.

Example #2 Determining Whether a Graph Defines a Function � Use the vertical line test to determine whether a graph represents a function. If not, give an input value that corresponds to more than one output value. Yes, it is a function. No, a vertical line passes through (0, 4) & (0, -4)

Analyzing Graphs Increasing Decreasing Always describe a graph from left to right. Constant

Example #3 Where a Function is Increasing/Decreasing �On what interval is the function increasing? decreasing? constant? Increasing: (-3, 0) Decreasing: None Constant: (- ∞, -3] [3, ∞)

Local Maxima & Minima While not always the highest or lowest point on a graph, they are for their neighborhood. Peaks or Local Maxima Valleys or Local Minima

Example #4 Finding Local Maxima/Minima �Find approximations for local maxima and minima using the graphing calculator. Finding local maximums and minimums with the graphing calculator are a lot like finding the intercepts. It asks you to specify a left and right bound around the local extreme as well as to guess its approximate location. This graph doesn’t properly show the local extrema so we must zoom in where the function dips.

Example #4 Finding Local Maxima/Minima �Find approximations for local maxima and minima using the graphing calculator. To zoom, press ZOOM 1: ZBox. This will allow you to create a box around the area you wish to magnify. Move the cursor to the upper left location of where you want to start the rectangle, press ENTER. Here both the local maximum and minimum are visible. Now move the cursor again, this time a rectangle is being drawn on the screen. When you have finished making the box around the region press ENTER again to zoom in.

Example #4 Finding Local Maxima/Minima �Find approximations for local maxima and minima using the graphing calculator. To find the local maximum, press 2 nd TRACE (CALC) 4: maximum. Specify the left and right bounds, I used 0 and 0. 76 which is a point in between the hill and valley. Guess a value in between, and the local maximum is where x ≈ 0. 304. For the minimum repeat this process but specify 3: minimum. I used 0. 76 and 2 for the bounds. The local minimum occurs where x ≈ 1. 096.

Concavity & Inflection Points inflection point concave up concave down If a segment connecting two points on an interval is above the curve, the interval is said to be concave up. If below the curve, then it is concave down. The point on a graph where concavity changes is called the point of inflection.

Example #5 Analyzing a Graph �Graph the function and find the following: A. All local maxima & minima x ≈ -2. 1547 and x ≈ 0. 1547 B. Intervals where the function is increasing and decreasing Increasing: (-∞, -2. 1547) (0. 1547, ∞) Decreasing: (-2. 1547, 0. 1547)

Example #5 Analyzing a Graph �Graph the function and find the following: C. Estimate any points of inflection x ≈ -1 D. Intervals where the function is concave up and concave down. Concave Down: Concave Up: (-∞, -1) (-1, ∞)

Example #6 Graphing a Piecewise-Defined Function �Graph the piecewise-defined function below: A piecewise function is made up of pieces of various functions. The dotted lines represent the entire graph of each function while the red parts are what are defined by the given intervals.

Example #6 Graphing a Piecewise-Defined Function �Graph the piecewise-defined function below: To graph these on the Y = screen, you must rewrite any compound inequalities for the domain as separate parts. Also put parentheses around the functions and the domains separately. To enter the inequality symbols on the Y = screen press 2 nd MATH (TEST)

Example #7 Graphing Absolute Value Functions �Rewrite and graph as a piecewise-defined function:

The Greatest Integer Function f(x) = [x] The greatest integer function (also called the floor function) outputs the greatest integer less than or equal to x. For example with f(x) = [x]: For negative values the function works like this: Notice each time the value is always rounded down to the integer, including negative inputs (remember a “bigger” negative value is a smaller number). The graph of the greatest integer function will be examined in the next example.

Example #8 The Greatest Integer Function �Graph the function The graph forms a step function.

Parametric Equations Parametric equations help to conveniently describe the equations of curves. The x and y coordinates of the points from a graph are given as a function of a third variable, t, called a parameter. A parametric graph can be thought of as representing the function: f(t) = (x, y) where x = x(t) & y = y(t) The butterfly curve formed by the parametric equations:

Example #9 Graphing a Parametric Equation �Graph the curve given by: t x=t– 2 y = t 2 + 3 t (x , y) – 3 – 2 = – 5 (– 3) 2+3(– 3) = 0 (– 3, 0) – 2 – 2 = – 4 (– 2) 2+3(– 2) = – 2 (– 2, – 2) – 1 – 2 = – 3 (– 1) 2+3(– 1) = – 2 (– 1, – 2) 0 0 – 2 = – 2 (0) 2+3(0) = 0 (0, 0) 1 1 – 2 = – 1 (1) 2+3(1) = 4 (1, 4) 2 2– 2=0 (2) 2+3(2) = 10 (2, 10)

Example #9 Graphing a Parametric Equation �Graph the curve given by: (x , y) (– 3, 0) (– 2, – 2) (– 1, – 2) (0, 0) (1, 4) (2, 10)

Example #9 Graphing a Parametric Equation �Graph the curve given by: Using the graphing calculator, these functions can be graphed in parametric mode. MODE Par Enter the function on the Y = screen using the variable key for the T. To get the graph to appear correctly press WINDOW and set the range for T from -10 to 10 and the Tstep to. 1

Writing an Equation in Parametric Form �To write the function y = f(x) in parametric form, let x=t y = f(t) �To write the function x = f(y) in parametric form, let x = f(t) y=t

Example #10 Graphing in Parametric Mode �Graph the following equations in parametric mode on the graphing calculator: A. B.