UNIT 5 RELATIONS FUNCTIONS AND GRAPHING FUNCTIONS A

UNIT 5 RELATIONS, FUNCTIONS, AND GRAPHING

FUNCTIONS • A relation is a set of ordered pairs. For example: {(3, 2), (4, 5), (6, 8), (7, 1)} • The relation can represent a finite set of ordered pairs or an infinite set.

Domain x-coordinate independent variable Range y-coordinate dependent variable

• The domain of a relation is the set of all xcoordinates from the ordered pairs in a relation. • The range of a relation is the set of all ycoordinates from the ordered pairs in a relation.

• A function is a special relation in which each member of the domain is paired with only one member of the range. • No two ordered pairs have the same first element

EXAMPLES • Determine whether each set of ordered pairs represents a function • {(3, 2), (4, 5), (6, 8), (7, 1)} • {(0, 5), (4, 3), (6, 5), (-7, -4)} • {(1, 3), (4, 2), (4, 1), (5, 6)} • {(6, -2), (11, -3), (14, 9), (-14, 11), (-14, 20), (21, -21)} • x 5 3 2 1 0 -4 -6 y 1 3 -2 2 -2

• Vertical line test – if a vertical line on a graph passes through more than 1 point it is not a function

• A solution to an equation or inequality in two variables is an ordered pair (x, y) that makes the equation or inequality true.

EXAMPLES – 3 x + 6 y = 12 (-4, 0) x + 5 y ≥ 11 (2, 1) 3 y – 5 x = 4 (-2, -2) 5 y < 3 x (-1, -3)

GRAPHING LINEAR FUNCTIONS Linear Equations • • Equations whose graphs are a straight line Find five values for the domain and make a table Plot each ordered pair Draw a straight line through the points Label the line with the original equation

EXAMPLE 1 x 2 x + 1 y (x, y)

EXAMPLE 2 x x - 4 y (x, y)

Standard Form of a Linear Equation where A, B, and C are real numbers, and A and B cannot both be zero.

EXAMPLE 3 • Graph 2 x + y = 5 x y (x, y)

WARM-UP •

HORIZONTAL & VERTICAL LINES y = c A _________ line parallel to the _____axis. x = c A ________line parallel to the____-axis. The y-coordinate has the The x-coordinate has the same value.

EXAMPLES

The cost of renting a car for a day is $64. 00 plus $0. 25 per mile. Let x represent the number of miles the car was driven and let y represent the rental cost, in dollars, for a day. a. Write an equation for the rental cost of the car in terms of the number of miles driven. b. Graph the equation

EXAMPLES • What must be the value of k be if (k, 4) lies on the line 3 x + y = 10? • What must the value of k be if (5, -3) lies on the line y – x = k • Find k such that (k, 5) is a solution of 3 x + 2 y = 22

SLOPE OF A LINE • Slope measures the steepness of a line by comparing the rise to the run. • The rise is the change in the “y” values while the run is the change in the “x” values. • Slope is often called the rate of change. • We represent slope with the letter m.

Positive Slope Negative Slope Zero Slope Undefined

SLOPE FORMULA

EXAMPLES • Find the slope of the line that is determined by the points (-2, 4) and (4, 2). • Find the slope of the line that goes through the points (3, -3) and (2, -3).

EXAMPLES •

Graph the line that has a slope of -3 and goes through the point (1, 3).

WARM-UP • Find k such that (k, 5) is a solution of 3 x + 2 y = 22 • Find the slope of the line that goes through the points (2, -4) and (-3, -3)

PARALLEL & PERPENDICULAR LINES • Parallel lines never intersect, therefore, the slopes of parallel lines are the same.

• Perpendicular lines intersect to form right angles. The slopes of perpendicular lines are negative reciprocals.

• When you multiply negative reciprocals, the product is -1. • When writing a negative reciprocal just think “flip and change the sign”.

GUIDED PRACTICE •

WARM - UP •

GRAPHING LINEAR EQUATIONS USING INTERCEPTS • The x intercept is the point at which a function crosses the x-axis. • The y intercept is the point at which a function crosses the y-axis. • If we know these two points, we can graph a line.

KEY POINTS • • Y-intercept: x value is 0; (0, y) X-intercept: y value is 0; (x, 0) To find the x-intercept, substitute 0 in for y and evaluate To find the y-intercept, substitute 0 in for x and evaluate

EXAMPLES

WARM-UP • If line A has a slope of 2 and line B is parallel to it, what is the slope of line B? • What are the x and y intercepts of the line with the equation y = 4 x – 2? • What does the graph of y = - 5 look like?

SLOPE INTERCEPT FORM •

EXAMPLES

WRITING EQUATIONS IN SLOPE INTERCEPT FORM • Given the slope and a point, we can write an equation in slope intercept form and then graph the line. • Method 1: • Substitute the x and y coordinate into y = mx + b • Evaluate to solve for “b” • “Put it all together” in slope intercept form

EXAMPLE 1 • Slope = ½ and goes through the point (2, -3)

EXAMPLE 2 • Write the equation of the line with slope of 2 that goes through the point (4, 6)

EXAMPLE 3 • Write the equation of the line that is parallel to y = 3 x-1 and goes through the point (0, 4)

EXAMPLE 4 • Write the equation of the line that is perpendicular to 7 x – 2 y = 3 and goes through the point (4, -1)

POINT SLOPE FORM •

EXAMPLE 5 • Write the equation of the line that has a slope of 4 and goes through the point (3, 5) then graph the line.

• If we are given two points, we can still write the equation: 1. Find the slope using the points given 2. Substitute the coordinates and slope 3. Evaluate 4. Equation should now be in slope intercept form

EXAMPLE 6 • Write the equation of the line that goes through the points (-3, -4) and (-2, -8).

EXAMPLE 7 • Write the equation of the line that goes through the points (2, 0) and (0, -1).