An Introduction to Functions LINEAR ALGEBRA

43210 In addition to level 3.0 and beyond what was taught in class, the student may:  Make connection with other concepts in math.  Make connection with other content areas. The student will understand and explain the difference between functions and non-functions using graphs, equations, and tables.  Compare properties of a function to a non- function. The student will be able to model and evaluate functions and non-functions.  Use graphs, equations, and tables to determine functions and non-functions. With help from the teacher, the student has partial success with level 2 and 3 elements. Even with help, students have no success with the functions. Focus 6 - Learning Goal #1: Students will understand and explain the difference between functions and non-functions using graphs, equations, and tables.

Why do we need to learn about functions? Functions allow us to make predictions. Functions can allow us to classify the data in our environment.

Speeding Rabbit Suppose a rabbit travels 256 feet in 4 seconds. Assume that this is a constant speed. Write a linear equation in two variables to represent the situation. In science you learned the relationship between distance and time. What is that equation? D = rt What values were given in this story that we could substitute into this formula? Distance = 256 feet Time = 4 secondsWe can solve for “r” rate.

D = rt 256 = r(4) Divide both sides by 4. r = 64 feet/second The rabbit’s personal equation to determine how far he could go in a certain amount of time is: d = 64t.

Use the equation to make predictions about the distance the rabbit traveled over various intervals of time. How can an equation help make predictions about distance the rabbit traveled in various amounts of time? We could make a table of values. The x column would correspond to time the rabbit ran. The y column would correspond to the distance the rabbit ran AFTER the time had passed. We could substitute the times into the rabbit’s formula D = 64t. Time (t)Distance (D) The equation (function) allows us to PREDICT the distance the rabbit has traveled after any duration of time.

A Function is… A function is a rule that assigns each input exactly one output. A rule is an equation. The equation for the rabbit’s distance after a specific amount of time was d = 64t. That equation is a function. Each input (time) had exactly one output (distance). This means at 3 seconds the rabbit had to have traveled 192 feet. The answer couldn’t have been 180, 192 and 300 feet. At 3 seconds the rabbit traveled 192 feet.

Determine if the following relations are functions: Yes. Each “x” value has only one “y” value. No. The “x” value 4 has two different “y” values.

Determine if the following relations are functions: 1.{(2, 5), (3, 9), (-4, 10), (6, -6), (-7, 5)} 2.{(3, 7), (-8, 2), (-1, 0), (3, 8), (2, 6)} Yes. Each “x” value has only one “y” value. No. The “x” value 3 has two different “y” values.

Walking Race: Mr. Furman and Mr. Ganey are in a walking race. Mr. Furman walks 18 meters in 6 seconds and Mr. Ganey walks 24 meters in 6.25 seconds. Find each teacher’s walking rate. D = rt Mr. Furman: 18 = r(6) Mr. Furman’s walking rate is 3 meters/second. Mr. Ganey: 24 = r(6.25) Mr. Ganey’s walking rate is 24/7 or 3.84 meters/second. Who will win the walking race?

Walking Race: Write a function to represent each teacher’s distance after any amount of time. Mr. Furman: D = 3t Mr. Ganey: D = 3.84t How long will it take Mr. Furman and Mr. Ganey to speed walk 150 meters? Substitute 150 meters into each equation and solve for time. Mr. Furman 150 = 3t = t It will take Mr. Furman 50 seconds to speed walk 150 meters. Mr. Furman 150 = 3t = t It will take Mr. Furman 50 seconds to speed walk 150 meters. Mr. Ganey 150 = 3.84t = t It will take Mr. Ganey 39 seconds to speed walk 150 meters. Mr. Ganey 150 = 3.84t = t It will take Mr. Ganey 39 seconds to speed walk 150 meters.