Drill 1) What quadrant would each point be located in: 2) If you start at the origin and go up 6 and then left 4 units what would be the coordinates of that point?

4.2 Relations Algebra 1

Relation A relation is a set of ordered pairs. Ex: {(2, 3) (3, 6) (-4, 8) (-11, 7)}

Relations Domain: the domain of a set is all the possible x-values. Range: the range of a set is all the possible y-values. Ex: {(2, 4) (3, 6) (8, 1) (-3, -4)} Domain is {2, 3, 8, -3} Range is {4, 6, 1, -4}

Mapping Mapping is when you write the domain in one oval and the range in another oval and then draw arrows from each value in the domain to it’s corresponding value in the range. 4 -1 5 8 6 7 10

Domain: {-3, 0, 2, 3, 4} Range: {-6, -4, 1, 2, 5}

Example What is the domain and range of this graph?

Drill {(3, 4) (-3, 6) (-8, 1) (-5, -4) (2, 4)} { (-5, 4) (3, 6) (4, 7) (-6, 9) (-5, 11)} Find the Domain and Range in each set and state whether or not each one is a function.

Inverse of a Relation To take the inverse of a relation you simply switch the x and y coordinates of each ordered pair. Ex: {(2, 4) (3, 6) (8, 1) (-3, -4)} Inverse is : {(4, 2) (6, 3) (1, 8) (-4, -3)}

DRILL What is the domain and range for this set? { (2, 3) (3, 8) (-2, 6) (5, 3) (4, 6) ( -2, -5)} 2) Which quadrant is each point in? (-3, 5) b) (-4, -5) c) (5, -2) 3) Solve for y if x = 3 y = 4x – 7

4.3 Equations and Relations

Using Equations As Relations When you have an equation and a given domain, you simply plug in the x-values to get the corresponding y-values. The y-values you get are your range. Ex: Solve the equation y = 3x + 4 if the domain is {-1, 2, 4, 5} X Y -1 2 4 5

y = -7x + 21 Examples Domain is: { -6, -2, 0, 1, 5, 11} Range is: { } 1 5 11 Domain is: { -6, -2, 0, 1, 5, 11} Range is: { }

5 minutes Warm-Up 1) Determine whether the point (0,3) is a solution to y = 5x + 3. 2) Graph y = -2x + 1

4.5 Linear Equations and Their Graphs Objectives: To graph linear equations in two variables

Linear Equations Equations whose graphs are lines are linear equations. Here are some examples: Linear Equations y = 3x + 7 6y = -2 9x – 15y = 7 Nonlinear Equations y = x2 - 4 x2 + y2 = 16 xy = 3

Example 1 Graph the equation 2x + 2y = 6. 2x + 2y = 6 solve for y -2x 3 2 2 1 2 y = 3 - x -2 5 y = 3 - (0) = 3 y = 3 - (1) = 2 y = 3 - (-2) = 5

Example 1 Graph the equation 2x + 2y = 6. x y 3 1 2 -2 5 8 2x + 2y = 6 4 x y 2 3 -8 -6 -4 -2 2 4 6 8 1 2 -2 -2 5 -4 -6 -8

Practice Graph these linear equations using three points. 1) 3y – 12 = 9x 2) 4y + 8 = -16x

Example 2 Graph the equation 3y – 6 = 9x. 3y – 6 = 9x solve for y +6 2 3 3 1 5 y = 3x + 2 -2 -4 y = 3(0) + 2 = 2 y = 3(1) + 2 = 5 y = 3(-2) + 2 = -4

Example 2 Graph the equation 3y – 6 = 9x. x y 2 1 5 -2 -4 8 2 -8 -6 -4 -2 2 4 6 8 1 5 -2 -2 -4 -4 -6 -8

Practice Graph these linear equations using three points. 1) 6x – 2y = -2 2) -10x – 2y = 8

6 minutes Warm-Up 1) Graph 4x – 3y = 12

Graphing Using Intercepts The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. The line shown intercepts the x-axis at (2,0). 8 6 4 We say that the x-intercept is 2. 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

Graphing Using Intercepts The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. The line shown intercepts the y-axis at (0,-6). 8 6 We say that the y-intercept is -6. 4 2 -8 -6 -4 -2 2 4 6 8 -2 -4 -6 -8

Example 1 Graph 4x – 3y = 12 using intercepts. *To find the y-intercept, let x = 0. 4x – 3y = 12 x y 4(0) – 3y = 12 -4 0 – 3y = 12 -3y = 12 -3 -3 y = -4

Example 1 Graph 4x – 3y = 12 using intercepts. *To find the x-intercept, let y = 0. 4x – 3y = 12 x y 4x – 3(0) = 12 -4 4x - 0 = 12 3 4x = 12 4 4 x = 3

Example 1 Graph 4x – 3y = 12 using intercepts. x y -4 3 8 6 4 2 -8 -6 -4 -8 -6 -4 -2 2 4 6 8 -2 3 -4 -6 -8

Example 2 Graph 2x + 5y = 10 using intercepts. *To find the y-intercept, let x = 0. 2x + 5y = 10 x y 2(0) + 5y = 10 2 0 + 5y = 10 5y = 10 5 5 y = 2

Example 2 Graph 2x + 5y = 10 using intercepts. *To find the x-intercept, let y = 0. 2x + 5y = 10 x y 2x + 5(0) = 10 2 2x + 0 = 10 5 2x = 10 2 2 x = 5

Example 2 Graph 2x + 5y = 10 using intercepts. x y 2 5 8 6 4 2 -8 -6 2 -8 -6 -4 -2 2 4 6 8 -2 5 -4 -6 -8

Practice Graph using intercepts. 1) 5x + 7y = 35 2) 8x + 2y = 24

Warm-Up 10 minutes Graph these equations: -x + 2y = 4 2x + 3y = 8

Linear Equations and Their Graphs Objectives: To graph linear equations that graph as horizontal and vertical lines

Graphing Horizontal and Vertical Lines The standard form of a linear equation in two variables is Ax + By = C, where A,B, and C are constants and A and B are not both 0. 3x + 4y = 12 6x + 7y = 23

Example 1 Graph y = -2. write the equation in standard form Ax + By = C (0)x + (1)y = -2 -8 -6 -4 -2 2 4 6 8 for any value of x y = -2

Example 2 Graph x = 7. write the equation in standard form Ax + By = C (1)x + (0)y = 7 -8 -6 -4 -2 2 4 6 8 x = 7 for any value of y

Practice Graph these equations. 1) x = 5 2) y = -4 3) x = 0

DRILL Find the next three terms in each pattern below. 1. 3, 6, 9, 12, … 2. 45, 39, 33, 27, … 3. 4, 5, 8, 13, 20, … 4. –12, –7, –2, 3, … 5. 1, 2, 4, 7, … 6. –2, –1.75, –1.5, –1.25, …

4.7 Arithmetic Sequences Objective:

Examples