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# 3.2

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### 3.2

1. 1. Graphs of FUNCTIONs 3.2
2. 2. A Function May be Defined by a Graph • “x” usually the independent or input variable • “y” usually the dependent or output variable • A graph is essentially a plot of inputs and their associated outputs • A point (x, y) on the graph can also be labeled as (x, f(x)) [remember y is the same as f(x)] • An open circle indicates the point is not part of the graph • A solid circle indicates the point is part of the graph
3. 3. Function Features • Vertical Line Test: If a vertical line intersects the graph no more than once, it’s a function! • Increasing function: the graph always rises as you move from left to right • Decreasing function: the graph always falls as you move from left to right • Constant function: the graph is horizontal
4. 4. Function Features • Local Maximum/Minimum: Peaks are local maximums, valleys are local minimums – TI 83/4: Play w/zoom and/or window size if necessary – CALC, 3: minimum or 4: maximum – Use arrows to select left bound, then right bound – Find min/max for f(x) = x3 – 1.8x2 + x +1
5. 5. Function Features • Concave Up: up = cup; if you connect two points the line segment is above the graph • Concave Down: down = frown; if you connect two points the line segment is below the graph • Inflection Points: a point where the graph changes concavity
6. 6. Graphs of Piecewise Functions • Combine the graphs of the formulas • Graph first formula as Y1, second part as Y2 • Inequalities found in TEST menu • Must use proper syntax Y1 = X2/(X≤1) Y2 = X + 2/((X>1)(X≤4)) • Calculator display will not show which endpoints are included or excluded f(x) = x+2 if 1< x ≤ 4 x2 if x ≤ 1
7. 7. Graph of Absolute Value Function • f(x) = |x| is a special case piecewise function • TI 83/4: MATH, NUM, 1:abs( - x if x < 0 f(x) = |x|= x if x ≥ 0
8. 8. Graph of Greatest Integer Function • For any number x, round down to the nearest integer less than or equal to x • Remember negative numbers round down, which is left on the number line! • TI-83/4: MATH, NUM, 5: int( • Graphs better in DOT mode vs CONNECTED • Easy to see why it is called a step function • Open circles are on the right side of each step f(x) = [x]
9. 9. Parametric Graphing • x and y are each a function of a third variable, t, which is called the “parameter” • The functions for x and y are called parametric equations – Note: x and y are functions of t, but y may or may not be a function of x • Parametric graph can be thought of as representing the function f(t) = (x,y) where x = x(t) and y = y(t) • TI-83/4: Select PAR mode (instead of FUNC)