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 = - 12

3 x – y = 123(0) – (-12) = 12

0 + 12 = 1212 = 12

True

Example 2 continued Let x = 1 and y = 9

3 x – y = 123(1) – (9) = 12

3 – 9 = 12-6 = 12False

Example 2 continued Let x = 2 and y = -6

3 x – y = 123(2) – (-6) = 12

6 + 6 = 1212 = 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

X Y 3x – y = 12

5 3 3(5) – 3 = 12

4 0 3(4) – 0 = 12

3 -3 3(3) – (-3) = 12

2 -6 3(2) – (-6) = 12

1 -9 3(1) – (-9) = 12

0 -12 3(0) – (-12) = 12Graph 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 variablesA linear equation in two variables is an

equation that can be written in the formAx + 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…

11500

10y x

Fill in the chart to determine ordered pairs

Products Sold (X)

0 100 200 300 400 1,000

Monthly Salary (Y)

Example 3 Use the graph of y = 1500 + 1/10 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) + 3 y = -2(2) + 3 y = -2(-1)

+ 3y = 3 y = - 1 y = 5

Example 4 Continued Graph each

ordered pairx y

0 3

2 -1

-1 5

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.

13

y x

Example 5 Continued

Fill in the table… using the equation

Graph the points

13

y x

x y

6 2

0 0

-3 -1

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 = x2

We know this is not a linear graph because the x2 term does not allow us to write it in the form Ax + By = C.

Step 1: Find ordered pair solutions

Example 6 continuedx y

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

Fill in the table using the equation:

y = xy = x22

Graph the ordered Graph the ordered pairspairs

Example 6 This curve is called a parabola.

Give it a try! Graph: y = -x2

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.

y x

Example 7 Continuedx y

-3 3

-2 2

-1 1

0 0

1 1

2 2

3 3

Fill in the table using the equation:

Graph the ordered Graph the ordered pairspairs

y x

Give it a try! Graph: 1y x