Functions, Equations, & Graphs Sections 2-1 to 2-4

Objectives To identify relations. To identify functions. Page 1

 Essential Understanding A pairing of items from two sets is special if each item from one set pairs with exactly one item from the second set.  A relation is a set of pairs of input and output values. You can represent a relation in four different ways

 The monthly average water temperature of the Gulf of Mexico in Key West, Florida varies during the year. In January, the mean (avg) water temperature is 69˚F, in February, 70˚F, in March, 75˚F, and in April, 78˚F. Represent this information in four different ways. MonthTemp (˚F) Jan. Feb. March April {(Jan., 69˚), (Feb, 70 ˚), (March, 75 ˚), (April, 78 ˚)} 69˚ 70˚ 75˚ 78˚ 69˚ 70˚ 75˚ 78˚ Jan Feb March April Month Temp (F˚) J F M A

 The domain of a relation is the set of inputs also called the x – coordinates, of the ordered pairs.  The range of the relation is the set of outputs, also called y - coordinates, of the ordered pairs. Problem 2Finding Domain and Range Use the relation from Problem 1 (skydivers) to find the domain and range of relation. Start by relisting the relation below. { (0, 10,000 ft), (4, 9744 ft), (8, 8976 ft), (12, 7696 ft), (16, 5904 ft)} The domain of the relation is: The range of the relation is: {0, 4, 8, 12, 16}{5904, 7696, 8976, 9744, 10,000} Got it?List the domain and range for the following relation: { (-3, 14), (0, 7), (2, 0), (9, -18), (23, -99) } The domain of the relation is: The range of the relation is: { -3, 0, 2, 9, 23 }{-99, -18, 0, 7, 14}

 Use { } to group the sets for both the domain and range. Separate the numbers with commas.  List the numbers in order from Least to Greatest  DO NOT REPEAT any number in either the domain or range that is used more than once. Ex. {(-1, 2) (-1, 3) (-4, 2) (0, 3) (1, 1)} Domain {-4, -1, 0, 1} Range {1, 2, 3}

 A function is a relation in which each element of the domain corresponds with exactly one element of the range.  Problem 3Identifying Functions Is the relation a function?  Got it?Is the relation a function? YES NO, 4 is repeated two times in the x spot. YES NO, 2 is repeated two times in the x spot.

 You can use the vertical - line test to determine whether a relation is a function.  The vertical - line test states that if a vertical line passes through more than one point on the graph of a relation, then the relation is NOT a function. HERE’S WHY IT WORKS If a vertical line passes through a graph at more than one point, there is more than one value in the range that corresponds to one value in the domain. NO YES NO

 Use the VLT to determine whether or not the relations pictured below are functions. NO YES

Objective To write and interpret Direct Variation Equations Page 5

 Some quantities are in a relationship where the ratio of corresponding values is constant. HINT: Direct variation is just a linear function with a “slope” and y-int equal to (0,0)  You can write a direct variation function as:  The formula says that, EXCEPT for (0,0), the ratio of all output-input pairs equals the constant k, called the constant of variation.  The constant of variation is just the slope of the line. Direct describes two quantities that vary in the same way: increasing together or decreasing together.

1. Find the constant of variation, k Use The “k” MUST be the SAME for all (x, y) pairs given to you!! If one pair has a different “k” value then the relation is NOT a direct variation. 2. If “k” is the same, then you MUST verify that (0, 0) is a point in the relation!! (Work backwards following the pattern you have found for the (x, y) pairs given to you.)

For each function, determine whether y varies DIRECTLY with x. If so, state the constant of variation and the function rule. Got it?

For each function, determine whether y varies directly with x. If so, state the constant of variation. Got it? Yes, Direct Function Constant is 7/3 No! Not Direct Function

 In a direct variation, is the same for all pairs of data where. So is true for the ordered pairs and, where neither nor is zero. Problem 3Using a Proportion to Solve a Direct Variation Suppose y varies directly with x, and and. What is y when ?

 The graph of a direct variation function is always a straight line through the origin. Problem 5Graphing Direct Variation Equations What is the graph of each direct variation equation?

Got it?Graphing Direct Variation Equations What is the graph of each direct variation equation?

 Write direct variation if the graph is a direct variation. Write linear if the graph is a linear function that is NOT a direct variation Yes, Direct No, doesn’t go through the origin – Linear Only y= -2xy=x y= -x + 2

To graph linear equations. To write equations of lines. Page 9

Consider a nonvertical line in the coordinate plane. If you move from any point on the line to any other point on the line, the ratio of the vertical change to the horizontal change is constant. That constant ratio is the slope of the line. TThe slope of a nonvertical line is the ratio of the vertical change to the horizontal change between two points.

Problem 1 Finding Slope What is the slope of the line that passes through the given points? Got it?What is the slope of the line that passes through the given points? a.(-3, 7) and (-2, 4)b.(3, 1) and (-4, 1)c.(7, -3) and (7, 1) a.(5, 4) and (8, 1)b.(2, 2) and (-2, -2)c.(9, 3) and (9, -4)

AA function whose graph is a straight line is a linear function. You can represent a linear function with a linear equation, such as y = 6x – 4. AA solution to a linear is equation is any ordered pair (x, y ) that makes the equation true.

A special form of a linear equation is called slope-intercept form. An intercept of a line is a point where a line crosses an axis. The y-intercept of a nonvertical line is the point at which the line crosses the y-axis. The x-intercept of a nonhorizontal line is the point at which the line crosses the x-axis. The slope-intercept form of an equation of a line is y = mx + b, where m is the slope of the line and (0, b) is the y-intercept.

a.What is an equation of a line where and the y-intercept is (0, -3)? b.What is the equation of the line? a. and the y-intercept is (0, 5)?b.What is the equation of the line? Problem 2Writing Linear Equations Got it?What is the equation of the line for each problem?

Problem 3Writing Equations in Slope-Intercept Form Write the equation in slope intercept form. State the slope and the y-intercept. Got it?Write the equation in slope intercept form. State the slope and the y-intercept.

To write an equation of a line given its slope and a point on the line. Page 13

 The slopes of two lines in the same plane indicate how lines are related.  Given the slope and y-intercept, you can write the equation of a line in slope-intercept form.  Given a point and slope of a line, you can write an equation in point-slope form.

A line passes through (-5, 2) with a slope of. What is the equation of the line? Got it? What is the equation of a line that passes through (7, -1) and has a slope of -3? Problem 1Writing an equation given a point and the slope

A line passes through (3, 2) and (5, 8). What is the equation of the line? Got it? A line passes through (-5, 0) and (0, 7). What is the equation of the line? Problem 2Writing an equation given two points

Another form of the equation of a line is standard form, in which the sum of the x and y terms are set equal to a constant. What is an equation of the line in standard form? Got it? What is an equation of the line in standard form? Problem 3Writing an equation in standard form

Problem 4Graphing an equation using intercepts What are the intercepts of ? Graph the equation. x-intercept y-intercept

Got it? What are the intercepts of ? Graph the equation. x-intercept y-intercept

Got it? The office manager of a small office ordered 140 packs of printer paper. Based on average daily use, she knows that the paper will last about 80 days. What graph represents this situation? What is the equation of the line in standard form? How many packs of printer paper should the manager expect to have after 30 days?

What is the equation, in slope-intercept form, of the line parallel to passing through (1, -3)? What is the equation, in slope-intercept form, of the line perpendicular to passing through (8, 5)? Problem 6Writing Equations of Parallel & Perpendicular Lines

What is the equation, in slope-intercept form, of the line parallel to passing through (4, -2)? What is the equation, in slope-intercept form, of the line perpendicular to passing through (0, 6)? Got it?

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 A piecewise function has different rules (or equations) for different parts of its domain (or x-values). Example 1Graph the piecewise function x < 0f(x) = -x x ≥ 0f(x) = x

 A step function pairs every number in an interval with a single value. The graph looks like steps.  One step function is the greatest integer function Example 3What is the graph of the function