FUNCTIONS What are they?

Goal Learn general properties that apply to all types of functions Transform a function so that its graph fits real-world data Obtain goals Graphically Algebraically Numerically Verbally

Overview (shooting arrow) Graphically  Algebraically: y = −4.9x2 + 20x + 5 Numerically: x (s) y (m) 0 5.0 1 20.1 2 25.4 3 20.9 Verbally: -4.9 is constant pertaining to gravitational acceleration, 20 is initial velocity, and 5 represents initial height of arrow.

Cup of Coffee How does the temperature of a cup of coffee relate to the time it sits??????? There is one and only one temperature at any one given time, sooooooooooo Temperature is a function of time!

Understanding Coffee Algebraically Numerically y = 20 + 70(0.8)x. x (min) y (°C) 0 90 5 42.9 10 27.5 15 22.5 20 20.8

Verbal Coffee Mathematical Model Function that is used to make predictions and interpretations about something in the real world Temperature depends on Time? Or time depend on Temperature? Temp is Dependent Variable Temp depends on time it has been cooling Time is Independent Variable Cannot change time by changing temp!

Domain/Range Domain: The set of values the independent variable of a function can have is called the Domain Range: The set of values of the dependent variable corresponding to the domain is called the range of the function The horizontal line at 20°C is called an asymptote. “Not due to coincide.” End behavior of the graph. ALWAYS PLOT INDEPENDENT VARIABLE ON X-AXIS AND DEPENDENT VARIABLE ON Y-AXIS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Example 1 The time it takes you to get home from a football game is related to how fast you drive. Sketch a reasonable graph showing how this time and speed are related. Give the domain and range of the function.

Final Graph Domain: 0 ≤ speed ≤ speed limit Range: time ≥ minimum time at speed limit