1.5 – Analyzing Graphs of Functions What you should learn: Use the vertical line test for functions. Find the zeros of functions Determine intervals on which functions are increasing or decreasing and determine relative maximum and relative minimum values of functions Determine average rate of change of a function Identify even and odd functions

1.5 – Analyzing Graphs of Functions Example 1: Use the graph of the function f to find The domain of the function The function values at f(-1) and f(2) The range of the function

1.5 – Analyzing Graphs of Functions Example 1b: Use the graph of the function f to find The domain of the function The range of the function #1

1.5 – Analyzing Graphs of Functions Example 1c: Use the graph of the function f to find The domain of the function The range of the function #4

1.5 – Analyzing Graphs of Functions Example 1d: Use the graph of the function to find the indicated function values f(-2) f(-1) f( ½ ) f(1) #5

1.5 – Analyzing Graphs of Functions Example 2: Use the Vertical Line Test to decide whether the graphs represent y as a function of x

1.5 – Analyzing Graphs of Functions Example 2: Use the Vertical Line Test to decide whether the graphs represent y as a function of x

1.5 – Analyzing Graphs of Functions Example 3: Find the zeros of each function. 𝑓 𝑥 =3 𝑥 2 +𝑥−10 𝑔 𝑥 = 10− 𝑥 2 ℎ 𝑡 = 2𝑡−3 𝑡+5

1.5 – Analyzing Graphs of Functions Increasing and Decreasing Functions:

1.5 – Analyzing Graphs of Functions Example 4: Use the graphs to describe the increasing and decreasing behavior or each function.

1.5 – Analyzing Graphs of Functions Example 4b: Use the graphs to describe the increasing and decreasing behavior or each function. # 34 and #35

1.5 – Analyzing Graphs of Functions The points at which a function is increasing, decreasing, or constant behavior are helpful in determining the relative maximum and relative minimum values of a function. (These are also sometimes called local max and local min)

1.5 – Analyzing Graphs of Functions Example 5: Use a graphing utility to approximate the relative max and relative min of the function: 𝑓 𝑥 =3 𝑥 2 −4𝑥−2

1.5 – Analyzing Graphs of Functions Example 5b: Use a graphing utility to approximate the relative max and relative min of the function: 𝑓 𝑥 =𝑥(𝑥−2)(𝑥+3)

1.5 – Analyzing Graphs of Functions Average Rate of Change: You know that the slope of the line can be interpreted as a rate of change. For a nonlinear graph whose slope change at each point, the average rate of change between any two points is the slope of the line through those two points. The line through the two points is called the secant line, and the slope of this line is denoted as 𝑚 𝑠𝑒𝑐 .

1.5 – Analyzing Graphs of Functions Example 6: Find the average rate of change of 𝑓 𝑥 = 𝑥 3 −3𝑥 from 𝑥 1 =−2 𝑡𝑜 𝑥 2 =0 from 𝑥 1 =0 𝑡𝑜 𝑥 2 =1

1.5 – Analyzing Graphs of Functions Example 7: The distance s (in feet) a moving car is from a stoplight is given by the function s 𝑡 =20 𝑡 3 2 , where t is the time (in seconds). Find the average speed of the car from 𝑡 1 =0 𝑡𝑜 𝑡 2 =4 from 𝑡 1 =4 𝑡𝑜 𝑡 2 =9

1.5 – Analyzing Graphs of Functions Even and Odd Functions: A function is said to be even if its graph is symmetric with respect to the y-axis and to be odd if its graph is symmetric with respect to the origin.

1.5 – Analyzing Graphs of Functions Example 8: Determine whether the function is even, odd, or neither. Then describe the symmetry. 𝑓 𝑥 = 𝑥 6 −2 𝑥 2 +3 𝑔 𝑥 = 𝑥 3 −5𝑥 𝑔 𝑠 =4 𝑥 2 3