WARM UP What is a function. How are they used.

FUNCTIONS

OBJECTIVES Work with functions that are defined algebraically, numerically, or verbally Understand and explain explain algebraic functions Review functions learned in Algebra 1 and Algebra 2. Graph functions

IMPORTANT TERMS & CONCEPTS Function Expressing mathematical ideas graphically, algebraically, numerically & verbally. Mathematical model Dependent variable Domain Range asymptote Extrapolation Interpolation

Functions If you pour a cup of coffee, it cools more rapidly at first, than less rapidly, finally approaching room temperature. Since there is one and only one temperature at any given time, the temperature is called a function of time. You can show the relationship between coffee temperature and time graphically. The graph shows the temperature y, as function of time x. At x = 0, the coffee has just been poured

Functions The graph shows that as time goes on, the temperature levels off, until it is so close to room temperature, 20 degree centigrade, that you can’t tell the difference. This graph might have come from an algebraic equation, From the equation, you can find numerical information. If you enter the equation into your grapher, then use the table feature, you can find these temperatures rounded to 0.1 degrees. x (min)y (°C)

MATHEMATICAL MODELS Functions that are used to make predictions and interpretations about something in the real world are called mathematical models. Temperature is the dependent variable because the temperature of the coffee depends on the time is has been cooling. Time is the independent variable. You cannot change time simply by changing coffee temperature. Always plot the independent variable on the horizontal axis and dependent variable on the vertical axis.

GRAPHING TERMS The set of values the independent variable of a function can have are called domain. In the cup of coffee example, the domain is the set of non-negative numbers or x > 0. The set of values of the dependent variable corresponding to the domain is called the range of the function. Time is If you don’t drink the coffee (which would end the domain) the range is the set of temperatures between 20 C and 90 C, including 90 degrees centigrade or 20 < x < 90. The horizontal line at 20 is called the asymptote. The graph gets arbitrarily close to the asymptote but never touches it.

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. Tell the domain and range of the functions. Solution: It seems reasonable to assume that the time it takes depends on the speed you drive. So you must plot time on the vertical axis and speed on the horizontal axis. To see what the graph should look like, consider what happens to the time as the speed varies. Pick a speed value and plot a point for the corresponding time.

EXAMPLE 1 CONTINUED Then pick a faster speed. Because time will be shorter, plot a point closer to the horizontal axis. For a slower speed, the time will be longer. Plot a point farther from the horizontal axis. Finally, connect the points with a smooth curve, since it is possible to drive at any speed within the speed limit. The graph never touches either axis, as shown. If the speed were zero, you would never get home. The length of time would be infinite. Also, no matter how fast you drive, it will always take some time to get there. You cannot arrive instantaneously. Domain: 0 < speed < speed limit Range: time > minimum time at speed limit