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INTRODUCTION TO FUNCTIONS • A function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on the input. o An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By substituting a number for x, an output can be determined. • A table can also be used to show input and output values. In this situation, the rule must be determined. • Word problems can also be used to describe situations that can be translated into functions. • The equation of a line, y = mx + b, is a function in which m is the slope of the line, b is the y-intercept, and x & y are the coordinates of a point on the line. The y-intercept is where a line crosses the y-axis. An equation of a line can be graphed by plugging in at least three numbers for x to produce the corresponding values for y. These are then the coordinates of three points on the line, which can be plotted and connected to form the line. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.

How to use introduction to functions • A function is a rule that tells what needs to be done to an input to produce an output. If a rule says to divide by 2 and add 1, then those are the operations that must be performed on the input. In this case if the inputs were the numbers, 2, 4, and 6, the output would be: Ex. Input Function Output 2 ÷ 2 + 1 2 4 ÷ 2 + 1 3 6 ÷ 2 + 1 4 • A function can also be noted as an equation, y = some operations performed on x, where x is the input and y is the output. o An example of this is y = x/4 - 2 or y = 8x + 1. A table can also show the input and output. For example: Given this table, what would the rule be? Each input is increased by 5, so the rule is add 5 or y = x + 5. • Graphs can also be used to determine output values and rules. To determine output values, follow the x-axis to the input value and go up until a point on the graph is reached, this is the output value. To determine a rule, write all the input and output values in a table as above and find a correlation between the two, this would be the rule. • The equation of a line, y = mx + b, has three parts; the slope, m; the y-intercept, b; and the point (x, y). If the slope and y- intercept are given, the equation of a line can be found by plugging in the slope for m and the y-intercept for b. o Ex. What is the equation of a line with a slope of 5 and a y-intercept of -2? y = mx + b, m = 5 and b = -2 → y = 5x - 2 input 1 2 3 4 5 6 output 6 7 8 9 10 11 © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.

• To find the slope of a line, the equation of a line must be in the form, y = mx + b. If an equation of a line is not in that form, it must be put into that form before the slope can be found. Ex. What is the slope of the line 5x + y = 3? Ex. 5x + y = 3 needs to change to y = -5x + 3 Once the equation is in the correct form, the slope is found, -5. • A line with a negative slope is a line that is going downhill. A line with a positive slope is a line that is going uphill. A line with a zero slope is a horizontal line and a vertical line has an undefined slope. The y coordinates can also be found with the equation of a line by plugging in the x values and evaluating. Try This! What is the output for the function, subtract 7, for the input 10, 11 and 12? What are the y values for the function, y = 2x -3, for the x values of 6, 8 and 10? What function does the table represent? What is the equation of a line with a slope of 2 and a y-intercept of 9? The equation of a line is y = -2 + 6, what is the slope? input 1 2 3 4 5 6 output 3 6 9 12 15 18 © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.