Relations, Functions, and Graphing 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 Range y-coordinate x-coordinate dependent variable independent variable Range y-coordinate dependent variable  

The domain of a relation is the set of all x-coordinates from the ordered pairs in a relation. The range of a relation is the set of all y-coordinates 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 -4 -6 y -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 –3x + 6y = 12 (-4, 0) 3y – 5x = 4 (-2, -2) x + 5y ≥ 11 (2, 1)

Graphing Linear functions 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 Linear Equations Equations whose graphs are a straight line

Example 1 x 2x + 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 2x + y = 5 x y (x, y)  

Warm-up Is it a function? {(6, -2), (11, -3), (14, 9), (-14, 11), (-14, 20), (21, -21)} Express 𝑥 −3≤𝑥<3 𝑤ℎ𝑒𝑟𝑒 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 in roster form.

Horizontal & Vertical Lines y = c x = c A _________________ line parallel to the _____- axis. A _______________line parallel to the____-axis. The y-coordinate has the same value. 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. Write an equation for the rental cost of the car in terms of the number of miles driven. Graph the equation

Examples What must be the value of k be if (k, 4) lies on the line 3x + 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 3x + 2y = 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 𝑆𝑙𝑜𝑝𝑒 𝐹𝑜𝑟𝑚𝑢𝑙𝑎: 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1  

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 Draw the line that goes through the point (-2,1) and has a slope of 3 2 .

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 3x + 2y = 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 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 1) 𝑦=2𝑥+3 2) − 1 2 𝑥−2=𝑦   3) 2𝑥−3𝑦=6 4) 𝑦=−4

Warm - Up Find the slope of the line that goes through the points (2, -4) and (-3, -3) Are the lines 𝑦=2𝑥+3 and − 1 2 𝑥−2=𝑦 perpendicular? Justify your answer. What can you tell me about the slopes of two lines that are parallel?

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 = 4x – 2? What does the graph of y = - 5 look like?

Slope intercept form 𝑦=𝑚𝑥+𝑏 We call the above equation slope intercept form because the m represents the slope and the b is the y-intercept. If we have the slope and the y-intercept, we can graph the line.

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 = 3x-1 and goes through the point (0,4)

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

Point slope form Method 2: Point Slope Form: 𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1 Substitute the x and y coordinates and slope into the equation Evaluate to get slope intercept form (y = mx + b)

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: Find the slope using the points given Substitute the coordinates and slope Evaluate 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).