Pre Calculus Section 1 2 Graphs of Functions
- Slides: 36
Pre. Calculus Section 1 -2 Graphs of Functions
Objectives v Find the domain and range of functions from a graph. v Use the vertical line test for functions. v Determine intervals on which functions are increasing, decreasing, or constant. v Determine relative maximum and relative minimum values of functions. v Identify even and odd functions.
The Graph of a Function v The graph of a function f is the collection of ordered pairs (x, f(x)) such that x is in the domain of f. v The geometric interpretations of x and f(x) are; v x: the directed distance from the y-axis v f(x): the directed distance from the x-axis f(x) The domain is {x| -2≤x<5} The range is {y| -4≤y≤ 4}. x
Domain & Range The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists. The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain. y Example: Find the domain and range of f (x) = x 2 – 2 x – 2 by investigating its graph. Range x The domain is the real numbers. The range is { y: y ≥ -3} or [-3, + ). 4 -4 Domain = all real numbers
Examples: Domain and Range from a Graph Calculate the Max. & Min. ∴ R = -6≤y≤ 6 or [-6, 6] Range Domain Calculate the zeros ∴ D = -3≤x≤ 3 or [-3, 3] Calculate the Min. ∴ R = y≥-3 or [-3, ∞) Range Domain The parabola continues to ∞ and covers the entire x-axis ∴ D = ℝ or (-∞, ∞)
y =|x| Your Turn Range Domain: ℝ or (-∞, +∞) Range: y≥ 0 or [0, +∞)
f(x) = (x + 2 2) Your Turn -4 Range Domain: ℝ or (-∞, +∞) Range: y≥-4 or [-4, +∞) Domain
, x≠ -3 Your Turn Domain: x>-3 or (-3, +∞) Range: y>-4 or (-4, +∞) Domain
f(x) = -(x + 3 2) -1 Your Turn Range Domain: ℝ or (-∞, +∞) Range: ℝ or (-∞, +∞) Domain
f(x) = x+2 2 -3 Your Turn Range Domain: ℝ or (-∞, +∞) Range: y>-3 or (-3, +∞) Domain
Determine the Domain and Your Turn Range for Each Function From Their Graph D = -5≤x≤ 9 or [-5, 9] R = -5≤y≤ 5 or [-5, 5]
Functions - Vertical Line Test v To determine whether or not a relation is in fact a function, we can draw a vertical line through the graph of the relation. v If the vertical line intersects the graph more than once, then that means the graph of the relation is not a function. v If the vertical line intersects the graph once then the graph shows that the relation is a function.
Examples: Vertical Line Test A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point. y y x Function One point of intersection y x Not a Function Two points of intersection
Your Turn Vertical Line Test: Apply the vertical line test to determine which of the relations are functions. y y x The graph does not pass the vertical line test. It is not a function. x The graph passes the vertical line test. It is a function.
Evaluating a Function Graphically Example: Function g’s graph is shown. Use the graph of g to evaluate g(– 1).
Evaluating a Function Graphically Solution (a) To evaluate g(-1) on the graph. (b) Find x = – 1 on the x-axis. Move upward to the graph of g. (c) Move across to the yaxis. Read the y-value: g(– 1) = 3.
Your Turn Find f(3) = -3
Your Turn Find f(7), f(3), and f(1) f(7) = 2 f(3) = 0 f(1) = DNE
Increasing and Decreasing Functions 1. Increasing function v The range values increase from left to right 2. v The graph rises from left to right v Positive slope Decreasing function v The range values decrease from left to right v 1. The graph falls from left to right 2. Negative slope To decide whether a function is increasing, decreasing, or constant on an interval, ask yourself “What does the graph do as x goes from left to right? ”
Graphs of Increasing and Decreasing Functions If could walk from left to right along the graph of an increasing function, it would be uphill. For a decreasing function, we would walk downhill.
Definition: Increasing, Decreasing, and Constant Functions Figure 7, pg. 2 -4
Increasing and Decreasing Functions v Increasing functions- rise from left to right v Decreasing functions- falls from left to right v Constant functions- flat (horizontal line) **A function that is increasing, decreasing, or constant, is described within its domain (in terms of x-values). ** Ask yourself: What does y do as x goes from left to right?
Intervals of Increase or Decrease We need to identify where the function is increasing or decreasing y -3 Increasing: x<1 or (- , 1) 1 5 x Decreasing: x>1 or (1, + )
Increasing, Decreasing, and Endpoints The concepts of increasing and decreasing apply only to intervals of the real number line and NOT to individual points. Decreasing: x<0 or (–∞, 0) Increasing: x>0 or (0, ∞) Do NOT say that the function f both increases and decreases at the point (0, 0). The point (0, 0) is the ‘turning point’.
Example Find the Intervals on the Domain in which the Function is Increasing, Decreasing, and/or Constant. Increasing: 3<x<5 Decreasing: x<-1 and x>5 Constant: -1<x<3
Your Turn v Determine the intervals over which the function is increasing, decreasing, or constant. Solution: Ask “What is happening to the y-values as x is getting larger? ”
Relative Minimum and Maximum Values of a Function A function f has a relative (local) maximum at x = c if there exists an open interval (r, s) containing c for all x between r and s. such that A function f has a relative (local) minimum at x = c if there exists an open interval (r, s) containing c for all x between r and s. such that Relative Maximums Relative Minimums
Relative Maximum Values of a Function Assuming the graph is continuous (no break) at the point where the function changes from increasing to decreasing, that point is called a relative maximum point.
Relative Minimum Values of a Function In the same manner, a relative minimum point occurs when the graph changes from decreasing to increasing.
Relative Minimum and Maximum Values of a Function **Maximum and Minimum values are based on y-values (or the output f(x)) of the function** v Relative Maximum (Peak) v Highest value in some open interval v Relative Minimum (Valley) v Lowest value in some open interval
Example: d. Relative Minimum Indicate and label each maximum or minimum point on the function graph, from left to right. e. Relative Maximum b. Relative Maximum s. Relative Minimum
Your Turn: Indicate and label each maximum or minimum point on the function graph, from left to right. b, Local c, & d Minimums e. Maximum r. Minimum s. Maximum
Your Turn: Find each maximum or minimum value and label as a maximum or minimum from left to right. f(6)=3 Relative Maximum f(4)=4 Relative Maximum f(2)=1 Relative Minimum f(5)=2 Relative Minimum
Use Graphing Calculator to Find Relative Minimum and Maximum Values of a Function v v v Example: v f(x) = -4 x 2 – 7 x + 3 v Answer: Max. (-0. 875, 6. 063) Example: v f(x) = -x 3 + x v Answer: Min. (-0. 58, -0. 38), Max. (0. 58, 0. 38) Your Turn: v F(x) = x 3 – 2 x 2 v Answer: Max. (0, 0), Min. (1. 3, -1. 2)
Even and Odd Functions A function f is called an even function if for all x in the domain of f. (Its graph is symmetric with respect to the y-axis. ) A function f is called an odd function if for all x in the domain of f. (Its graph is symmetric with respect to the origin. ) To determine odd or even function, calculate f(-x). If f(-x)=f(x) even, if f(-x)=-f(x) odd, and if f(-x)≠f(x) & f(-x)≠-f(x) neither. Example Decide if the functions are even, odd, or neither v v
Your Turn 1) Determine whether f(x) = 5 – 3 x is even, odd, or neither. v Answer: Neither 2) Determine whether f(x) = - x 2 + 3 is even, odd, or neither. v Answer : Even
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