8th Grade Common Core Math Booklet 5 Functions Part 2
One of the Main Idea of Functions: Use functions to model relationships between quantities What are functions? Functions are like machines. You give them an input and they give you an output. A function is essentially a rule that is assigned to an input to receive an output. Functions are written like this f(x) = 4x where f is the function name, (x) is the input value and 4x represents the rule to x. Any rule to x can be used. The rule is what the function does to x. Examples of Functions: f(x) = 6x - 5 g(x) = 2x + 3 f(x) = x / 2 Notice in the above examples that the letter g was used as the name of the function. Other variables can be used to express a function name, but f is the most common.
8 th Grade Common Core Math Standards: Standard 8.F.B.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. What the student learns: Students learn to find the slope and y-intercept from tables, graphs, or equations and learn how to write a function (equation) from that data. Standard examples: Find the slope and y-intercept from equation, table, and graph below: Example: y = 4x + 3 Answer: This equation is written in the form y = mx + b where m is the slope and b is the y-intercept. The slope in this equation is 4 and the y-intercept is 3
Example: Find the slope and y-intercept from this table: x y 0 3 2 4 8 7 Answer: The y-intercept can be found by looking at the y-value when x is 0. So in this case the y-intercept is 3.!!!"#$!"!!!!!! Slope can be found by using which is so if we use!!!"#$!"!!!!!! points form our table we see!!! is the slope.!!! The slope is!! which simplifies to!! The equation of this table is y =!! x + 3 Example: Find the slope and y-intercept from this table: 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 Answer: In this graph, the y-intercept is 1 because that is the point where it passes the y axis. The slope is!!!"#$!"!!!!"#$!"! points on the graph. Let s use (1,3) and (2,5). which can be found by using any two coordinate The slope would be!!!!!!!!!! which is!!!!!! which is!! or 2. The equation of the line must be written as y = mx + b where m is the slope and b is the y-intercept. So the equation of this line is y = 2x + 1
Standard 8.F.B.5: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. What the student learns: If a student is given a graph of a situation, students should be able to describe what is happening. If words describing a graph are given, students should be able to draw a graph that resembles the situation told. Standard examples: Describe a story for this graph for Josh (Blue) and Matthew (Green) who are reading the same book. 6 Chapters Read per Day 5 4 3 2 1 0 0 1 2 3 4 5 6 Possible Answer: Day Number Two boys, Josh and Matthew are reading the same book. Josh likes the book and reads two chapters on day one. Matthew doesn t like it that much and reads one chapter on day one. Josh enjoys the book for the next 3 days (days 2, 3, 4) and reads 4 chapters each day. In the same time Matthew enjoys the book more than before and reads 2 chapters on day 2, 4 chapters on day 3, and 6 chapter on day 4. Josh starts to think it is boring and reads 3 chapters on day 5. Matthew still thinks it is great on day 5 and reads another 6 chapters.
Example 2 Make a graph that: Stays the same for more than one day. Then it shoots up rapidly in one day. Next, it goes up a little for one day and then goes back down the next day where it stays the same for the last day Possible Answer: 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 Days There are many possible answers that students could come up with. This is just one example.
WHY THIS IS IMPORTANT Functions are used all the time in the real world and you may not even know you are using them. If you were to get a rental car for one week and you find two car rental places, you could use functions to find which place gives you the better deal. For example, at Place 1, it costs $180 a week to rent a car, but you get unlimited miles. Place 2 costs $156 a week but it costs an extra 12 per mile driven. If you graph each function, you can see which car is the better deal if you estimate or know the amount of miles you will be driving.