Graphing Equations Chapter 3.1

Objectives Plot ordered pairs Determine whether an ordered pair of numbers is a solution to an equation in two variables. Graph linear equations. Graph non-linear equations.

Important Vocabulary Plotted – located or graphed Ordered pair – represented by the notation (x,y) X – coordinate – associated with the x- axis Y – coordinate – associated with the y- axis

The Cartesian Coordinate System

Ordered Pairs Why are the points in a rectangular coordinate system called ordered pairs? ***Each ordered pair corresponds to exactly one point in the real plane and each point in the plane corresponds to exactly one ordered pair.***

Example 1 – Plotting points Plot each ordered pair: a.(2,1) b.(0,5) c.(-3,5) d.(-2,0) e.(-1/2, -4) f.(1.5, 1.5) A BC D E F

Give it a try! Plot each ordered pair: a. (3, -2) b. (0,3) c. (-4,1) d. (-1,0) e. (-2 ½, -3) f. (3.5, 4.5)

Concept Check Which of the following best describes the location of the point (3,-6) in a rectangular coordinate system? A. 3 units to the left of the y-axis and 6 unites above the x-axis B. 3 units above the x-axis and 6 units to the left of the y-axis C. 3 units to the right of the y-axis and 6 units below the x-axis D. 3 units below the x-axis and 6 units to the right of the y-axis

Solutions Solutions of equations in two variables consists of two numbers that form a true statement when substituted into the equation. A convenient notation for writing these numbers is as ordered pairs. If the solution contains variables x and y write them as a pair of numbers in the order (x, y) If any other variable is used, write them in alphabetical order.

Example 2: Determine whether (0,-12), (1,9), and (2, -6) are solutions of the equation 3x- y =12 Step 1: Substitute in each x value for x and each y value for y to determine if the ordered pair is a solution.

Example 2 continued Let x = 0 and y = x – y = 12 3(0) – (-12) = = = 12 True

Example 2 continued Let x = 1 and y = 9 3 x – y = 12 3(1) – (9) = 12 3 – 9 = = 12 False

Example 2 continued Let x = 2 and y = -6 3 x – y = 12 3(2) – (-6) = = = 12 True

Example 2 Continued Thus, (1,9) is not a solution of 3x – y = 12, but both (0,-12) and (2, -6) are solutions. In fact, the equation 3x – y = 12 has an infinite number of ordered pair solutions. Since it is impossible to list all solutions, we visualize them by graphing them.

Example 2 Continued XY3x – y = (5) – 3 = (4) – 0 = (3) – (-3) = (2) – (-6) = (1) – (-9) = (0) – (-12) = 12 Graph on board

Linear Equation The equation 3x – y =12 is called a linear equation in two variables, and the graph of every linear equation in two variables is a line. Linear Equations in two variables A linear equation in two variables is an equation that can be written in the form Ax + By = C Where A and B are not both 0. This form is called standard form.

Give it a try! Determine whether (0,-6), (1,4), and (- 1,-4) are solutions of the equation 2x + y = -6

Standard Form A linear equation is written in standard form when all of the variable terms are on one side of the equation and the constant is on the other side.

Real – Life Linear Equations Suppose you have a part-time job in a store that sells office supplies. Your pay is $1500 plus 10% or 1/10 of the price of the products you sell. If we let x represent products sold and y represent monthly salary, the linear equation that models your salary is …

Fill in the chart to determine ordered pairs Products Sold (X) ,000 Monthly Salary (Y)

Example 3 Use the graph of y = / 10 x to answer the following questions. a. If the salesperson has $800 of products sold for a particular month, what is the salary for that month? b. If the salesperson wants to make more than $1600 per month what must be the total amount of products sold?

Graph Graph the line and then find the corresponding salary for $800 products sold. You can also substitute $800 for x and solve for y. Find the corresponding point for $1600 salary on the graph.

Give it a try! Use the graph from Example 3 … A. If the salesperson sells $700 of products for a particular month, what is the salary? Find the total amount of products needed to be sold to make more than $1550 per month.

Example 4 Graph the equation: y = -2x + 3 Step 1: Choose three values for x Let ’ s say x = 0, 2, and -1 to find our three ordered pair solutions Step 2: Plug in each value for x and solve for y y = -2(0) + 3y = -2(2) + 3y = -2(-1) + 3 y = 3y = - 1y = 5

Example 4 Continued Graph each ordered pair xy

Intercept Notice that the graph crosses the y – axis at the point (0,3). This point is called the y-intercept. This graph also crosses the x – axis at the point (1.5, 0). This point is called the x-intercept

Finding x – and y - intercepts To find an x-intercept, let y = 0 and solve for x To find a y-intercept, let x = 0 and solve for y

Give it a try! Graph: y = 4x - 3

Example 5 Graph the linear equation Step 1: Choose x- values (To avoid fractions, we choose x-values that are multiple of 3!) Choose 6, 0, -3 Step 2: Substitute in the x values and solve for y.

Example 5 Continued Fill in the table … using the equation Graph the points xy

Give it a try! Graph y = -5x

Non-linear equations Not all equations in two variables are linear equations, and not all graphs of equations in two variables are lines. Remember linear equations are written in the form … Ax + By = C

Example 6 Graph: y = x 2 We know this is not a linear graph because the x 2 term does not allow us to write it in the form Ax + By = C. Step 1: Find ordered pair solutions

Example 6 continued xy Fill in the table using the equation: y = x 2 y = x 2 Graph the ordered pairs Graph the ordered pairs

Example 6 This curve is called a parabola.

Give it a try! Graph: y = -x 2

Example 7 Graph the equation: We know this is not a linear equation and its graph will not be a line. Since we do not know the shape of this graph we need to find many ordered pair solutions. We choose x – values and substitute to find corresponding y - values.

Example 7 Continued xy Fill in the table using the equation: Graph the ordered pairs Graph the ordered pairs

Give it a try! Graph: