Vocabulary Dependent Variable Independent Variable Input Output Function

Vocabulary • • • Dependent Variable Independent Variable Input Output Function Linear Function

Definition • Dependent Variable – A variable whose value depends on some other value. – Generally, y is used for the dependent variable. • Independent Variable – A variable that doesn’t depend on any other value. – Generally, x is used for the independent variable. • The value of the dependent variable depends on the value of the independent variable.

Independent and Dependent Variables On a graph; the independent variable is on the horizontal or x-axis. the dependent variable is on the vertical or y-axis. y dependent x independent

Example: Identify the independent and dependent variables in the situation. A veterinarian must weight an animal before determining the amount of medication. The amount of medication depends on the weight of an animal. Dependent: amount of medication Independent: weight of animal

Your Turn: Identify the independent and dependent variable in the situation. A company charges $10 per hour to rent a jackhammer. The cost to rent a jackhammer depends on the length of time it is rented. Dependent variable: cost Independent variable: time

Your Turn: Identify the independent and dependent variable in the situation. Camryn buys p pounds of apples at $0. 99 per pound. The cost of apples depends on the number of pounds bought. Dependent variable: cost Independent variable: pounds

Definition • Input – Values of the independent variable. – x – values – The input is the value substituted into an equation. • Output – Values of the dependent variable. – y – values. – The output is the result of that substitution in an equation.

Function • In the last 2 problems you can describe the relationship by saying that the perimeter (dependent variable – y value) is a function of the number of figures (independent variable – x value). • A function is a relationship that pairs each input value with exactly one output value.

Function You can think of a function as an input-output machine. input x 26 function y = 5 x 5130 x 0 output

Helpful Hint There are several different ways to describe the variables of a function. Independent Variable Dependent Variable x-values y-values Input Output Domain Range x f(x)

A function is a set of ordered pairs (x, y) so that each x-value corresponds to exactly one y-value. Function Rule Output variable Input variable Some functions can be described by a rule written in words, such as “double a number and then add nine to the result, ” or by an equation with two variables. One variable (x) represents the input, and the other variable (y) represents the output.

Linear Function • Another method of representing a function is with a graph. • A linear function is a function whose graph is a nonvertical line or part of a nonvertical line.

Example: Representing a Linear Function A DVD buyers club charges a $20 membership fee and $15 per DVD purchased. The table below represents this situation. Number of DVDs purchased x 0 1 2 3 4 5 Total cost ($) y 20 35 50 65 80 95 +15 +15 +15 Find the first differences for the total cost. constant linear Since the data shows a ______ difference the pattern is _____. If a pattern is linear then its graph is a straight _____. line

Relations ØA relation is a mapping, or pairing, of input values with output values. Ø The set of input values is called the domain. Ø The set of output values is called the range.

Domain & Range Domain is the set of all x values. Range is the set of all y values. Example 1: {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Domain- D: {1, 2} Range- R: {1, 2, 3}

Example 2: Find the Domain and Range of the following relation: {(a, 1), (b, 2), (c, 3), (e, 2)} Domain: {a, b, c, e} Range: {1, 2, 3} Page 107

Every equation has solution points which satisfy the equation). (points 3 x + y = 5 Some solution points: (0, 5), (1, 2), (2, -1), (3, -4) Most equations have infinitely many solution points. Page 111

Ex 3. Determine whether the given ordered pairs are solutions of this equation. (-1, -4) and (7, 5); y = 3 x -1 The collection of all solution points is the graph of the equation.

3. 3 Functions • A relation as a function provided there is exactly one output for each input. • It is NOT a function if at least one input has more than one output Page 116

In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT (DOMAIN) FUNCTION MACHINE Functions OUTPUT (RANGE)

Example 6 Which of the following relations are functions? R= {(9, 10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates).

Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Domain = {-3, 1, 3, 4} Range = {3, 1, -2} Function? Yes: each input is mapped onto exactly one output

Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 4 Domain = {-3, 1, 4} Range = {3, -2, 1, 4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1

The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117

Use the vertical line test to visually check if the relation is a function. (-3, 3) (4, 4) (1, 1) (1, -2) Function? No, Two points are on The same vertical line.

Use the vertical line test to visually check if the relation is a function. (-3, 3) (1, 1) (3, 1) (4, -2) Function? Yes, no two points are on the same vertical line

Examples Ø I’m going to show you a series of graphs. Ø Determine whether or not these graphs are functions. Ø You do not need to draw the graphs in your notes.