Relations and Functions 2.1

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1 Relations and Functions A 2 B D -5 5 E -2 C F -4 Relation a set of ordered pairs (Domain, Range). Mapping shows how each number of the domain is paired with each member of the range. Example 1 (2, 4), (3, 0), (5, -2), (6, 0) Function a special type of relation in which each element of the domain is paired with exactly one element in the range. Discrete Function set of individual points. Continuous Function the graph of a line or smooth curve. Example 2 Which, if any, relation is considered a function?

Vertical Line Test a vertical line used to determine if a relation is a function. If the line passes through no more than one point, then the relation is a function.

2 Vertical Line Test a vertical line used to determine if a relation is a function. If the line passes through no more than one point, then the relation is a function. Determine whether it is a function. Example 3 Example 4 Graph y = 3x + 2 State the domain and range. Example 5 Graph x = y + 2 State the domain and range. 2 Example 6 Given the function 2 g( x) = x - 4 and f(x) = 3x, find each value. a. g(3) b. g(-4) c. f(-1) Pg 68,16-48

3 Linear Equations 2.2 CD s cost $15.99 Write an equation that would find the total for x number sold: Since, y depends on x, y is considered the dependant variable and x is considered the independent variable. Linear Equation graph of a line. Standard Form (line) Ax + By = C x and y on the same side of equal sign No Fractions Write each in standard form. Example 1 Example 2 Example 3 3y = x y = 4 x + 5 2x + 4y = 10 Linear Function f(x) = mx + b No variable can have an exponent other that 1 x and y must be separated by + or - sign Constant Function m = 0 Horizontal Line State whether each function is a linear function.

4 Example 4 Example 5 f(x) = -3x 2 g( x) = x + 5 Example 6 Example 7 y = x + 5 x + xy = 1 Graph by the x and y intercept. Example 8 Example 9 5x + 2y = 10 3y = 5x - 10 Pg 76, evens

5 Slope 2.3 Slope (m) ratio of the change in vertical units to the change in horizontal units. rise vert. change y y - y m = = = = run hor. change x x - x Find the slope. Example 1 Example 2 (5, -1), (4, -7) (-2, 3), (-4, 4) Example 3 Example 4 (7, 5), (-3, 5) (5, 7), (5, -3) Example 5 Graph the line passing through (4, 5) with a slope of 3. Example 6 Graph a line passing through (1, 3) with a slope of 3. 4

6 Positive Slope Negative Slope 0 slope No Slope Parallel Lines same slope Perpendicular Lines negative reciprocal slopes. Example 7 Graph a line through (3, 2) that is parallel to a line with a slope of -3. Example 8 Graph a line through (2, -5) that is perpendicular to a line with a slope of -4. Example 9 Find the slope of 2x + 3y = 12 Pg 84, odds

7 Writing Linear Equations 2.4 Slope-Intercept Form y = mx + b m = slope b = y-intercept Example 1 Find the slope and y-intercept of 2x + 3y = 8 Example 2 Find the slope-intercept form of an equation of the line that has a slope of 2 and passes through (-1, 3). Point-Slope Form y - y = m( x - x ) 1 1 m = slope ( x, y ) = point 1 1 Example 3 Find an equation of the line that passes through (2, 3) and (1, 5).

8 Example 4 Write and equation of a line that passes through (3, 2) and is perpendicular to the line whose equation is y = 2x + 5. Example 5 Write and equation of a line that passes through (-6, 1) and m = no slope. What if m = 0? Pg 91, odds

9 Statistics Using Real World Data 2.5 Scatter Plot used when real-life data is collected, the points graphed usually do not form a straight line. Best-Fit Line a line used to approximate a linear relationship. Prediction Equation an equation used to predict outcomes by using the best-fit line. Example 1 The city of Whitsville s population for the past 10 years is listed below. Draw a scatter plot and find a prediction equation. Age of Whitsville Population 34 22, , , , , ,095

Example 2 Each of the seven executives oversees a varied number of salespersons. Below is a chart with the number of salespersons and total sales for one month for each executive.

10 Example 2 Each of the seven executives oversees a varied number of salespersons. Below is a chart with the number of salespersons and total sales for one month for each executive. Find the prediction equation for this relationship and predict the total sales for the seventh executive. # of salespersons Sales ? Graphing Calculator 1. Stat, Edit x in y in L L Stat, Calc, #4 (Linear Regression) L, L 1 3. y = 2 4. Vars, #5 (Statistics), EQ, enter 5. Graph. Pg 98, 3-12 skip 5

11 Special Functions 2.6 Direct Variation when a linear function in the form y = mx + b has b = 0 and m 0. Constant Function Identity Function Step Function Example 1 The price of aluminum given by a recycling center is based upon weight. If the aluminum weighs more than 0 pounds but less than or equal to 1 pound, there is no payment. If the aluminum weighs more than 1 pound and less than or equal to 2 pounds, the price is $2.00. For each additional pound, the price increases $1.00. Graph the function. Absolute Value Function Example 2 Example 3 Example 4 y = x y = x + 2 y = x + 2

12 Greatest Integer Function [ x ] = means greatest integer not greater than x Example 5 Example 6 [8.7] = [-1.6] = Example 7 Graph y = [x] + 2 Pg 106, 7-35 odds, skip 33

13 Graphing Inequalities 1. >, <, 2. Test a point in both regions. 3. Shade the region with a true inequality. Linear Inequalities 2.7 Example 1 Example 2 Graph y > 2x 4 Graph 3y 2x 6 Example 3 Example 4 Graph y < 3 Graph 4y x > 8 Example 5 Example 6 Graph 2x - 3y < 6 Graph y < x + 4 Pg 112, 2-27

Section 18-1: Graphical Representation of Linear Equations and Functions

Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as