graphing equations chapter 3.1. objectives plot ordered pairs determine whether an ordered pair of...
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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