Presentation on theme: "5.7 Describe and Compare Function Characteristics How do you analyze and compare functions? How do you sketch a graph from a verbal description?"— Presentation transcript:
5.7 Describe and Compare Function Characteristics How do you analyze and compare functions? How do you sketch a graph from a verbal description?
Increasing over an intervalDecreasing over an interval Rises from left to rightFalls from left to right
FLYING Sketch a graph of a(t) where a represents altitude and t represents time for the situation below. Label key information such as local minima and maxima and intervals where a(t) is increasing or decreasing. A small plane flying at a constant cruising altitude is caught in a storm. Air currents carry the plane up before pushing it down rapidly below its original altitude. The plane exits the storm and returns gradually to its cruising altitude.
1. Sketch a graph for the situation described below. Label key information. The US gross domestic product (GDP) growth rate over a 2 year period can be modeled by a polynomial that began slightly positive before dropping to significantly negative, swinging back to significantly positive, then falling to a growth rate a little higher than at the beginning of the period.
Average Rate of Change For a linear function, the rate of change is constantthe slope. For a nonlinear function, you can find the average rate of change over any interval by finding the slope of the segment that connect the points on the graph determined by the endpoints of the interval.
SOLUTION For the function f(x) = 2 x – 2 + 1, find the average rate of change over the intervals [–2, 0], [0, 2], [2, 4], [4, 6], and [6, 8]. What happens to the average rate of change as x increases? What does this mean for the graph of f(x) ? The average rate of change over an interval [x 1, x 2 ] is. f(x 2 ) – f(x 1 ) x 2 – x 1 f(x) = 2 x – 2 + 1
The average rate of change is always positive, so the graph always rises. The average rate of change increases over successive intervals of the domain (by a factor of 4 for each increase of 2 in x ). Because the average rate of change over each interval is equivalent to the slope of the segment from (x 1, f(x 1 )) to (x 2, f(x 2 )), the graph rises more and more steeply as x increases.
FOOTBALL Fantasy football participants draft players for their teams. Below are models predicting fantasy points for two positions as a function of the players rank. Compare fantasy points as a function of rank for the models. Tight Ends Running Backs Comparing ordered pairs, the points for running backs are always greater than for tight ends for each rank over the domain shown, but for running backs, the points decrease more rapidly as rank changes than for tight ends.
Determine whether the function is even, odd, or neither. a. f(x) = x 3 – 7x b. g(x) = 6 x + 1 Replace x with –x in the equation for the function, and then simplify. a. f(–x) = (–x) 3 – 7(–x) = –x 3 + 7x = –(x 3 – 7x) = – f(x) Because f(–x) = – f(x), the function is odd. b.g(–x) = = 6 (–x) –x + 1 Because and –, g(x) is neither even nor odd. 6 –x x –x x + 1