Graphing Linear Equations

Graphing Linear Equations

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Table of Contents

• Vocabulary Review

• Defining Slope on the Coordinate Plane• Tables and Slope

• Tables

• Slope Formula

• Slope & y-intercept

• Slope Intercept Form• Rate of Change• Proportional Relationships and Graphing• Slope and Similar Triangles• Parallel and Perpendicular Lines• Solve Systems by Graphing• Solve Systems by Substitution• Solve Systems by Elimination• Choose Your Strategy• Writing Systems to Model Situations

Day 1Points &

Coordinates

y-axis: vertical number line that extends indefinitely in both directions from zero. (Up- positive Down- negative)

x-axis: horizontal number line that extends indefinitely in both directions from zero. (Right- positive Left- negative)

Origin: the point where zero on the x-axis intersects zero on the y-axis. The coordinates of the origin are (0,0).

II I

III IV

Vocabulary ReviewCoordinate Plane: the two dimensional plane or flat surface that is created when the x-axis intersects with the y-axis. Also known as a coordinate graph and the Cartesian plane.

Quadrant: any of the four regions created when the x-axis intersects the y-axis. They are usually numbered with Roman numerals.

To graph an ordered pair, such as (4, 8), you start at the origin (0, 0)and then go left or right on the x-axis depending on the first number and then up or down from there parallel to the y-axis.

So to graph (4,8), we would go 4 to the right and up 8 from there.

Linear Equation:

Any equation whose graph is a line.

One way to check this is to create a table of values.

Tables

Return to Table

of Contents

Geometry Theorem:

Through any two points in a plane there can be drawn only one line.

Given y = 3x + 2, we want to graph our equation to show all of the ordered pairs that make it true.

So according to this theorem from Geometry, we need to find 2 points.

One way is to create a table of values.

Let's consider the equation y= 3x + 2.

We need to find pairs of x and y numbers that make equation true.

Let's find some values for y = 3x + 2.

Pick values for x and plug them into the equation, then solve for y.

x 3(x)+2 y (x,y)

0 3(0)+2 2 (0,2) 2 3(2)+2 8 (2,8)

-3 3(-3)+2 -7 (-3,-7)

x 3(x) + 2 y (x,y)

0 3(0) + 2 2 (0,2) 2 3(2) + 2 8 (2,8)

-3 3(-3) + 2 -7 (-3,-7)

Now let's graph those points we just found.

Notice anything about the points we just graphed?

That's right! The points we graphed form a line.

The theorem says we only needed 2 points, so why did we graph 3 points?

The third point serves as a check.

Graph y = 2x+4

x 2x+4 y (x,y)0 2(0)+4 4 (0,4)3 2(3)+4 10 (3,10)-1 2(-1)+4 2 (-1,2)

Now graph your pointsand draw the line.

x 2x+4 y (x,y)

click for table

x -2(x)+1 y (x,y)0 -2(0)+1 1 (0,1)3 -2(3)+1 -5 (3,-5)-1 -2(-1)+1 3 (-1,3)

Graph y = -2x+1

Now graph your pointsand draw the line.

x -2(x)+1 y (x,y)

click for table

x ¾(x)-3 y (x,y)0 ¾(0)-3 -3 (0,-3)4 ¾(4)-3 0 (4,0)-4 ¾(-4)-3 -6 (-4,-6)

Graph y = ¾x 3

Now graph your pointsand draw the line.

x ¾(x)-3 y (x,y)

Recall that in the previous examplethat even though the number in frontof x was a fraction, our answers wereintegers.

x ¾(x)-3 y (x,y)0 ¾(0)-3 -3 (0,-3)4 ¾(4)-3 0 (4,0)-4 ¾(-4)-3 -6 (-4,-6)

Notice the x-values chosen are zero, the denominator and the opposite of the denominator.Why?

Move me to find out!

1 Given an equation of y = 2x - 5, what is y when x = 0?

2 Given an equation of y = 2x - 5, what is y if x is 1/2?

A -5B -4C -3D 2E 4

3 Which point is on the line 4y − 2x = 0?A (−2,−1)B (−2,1)C (−1,−2) D (1,2)

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

4 Which point lies on the line whose equation is 2x − 3y = 9?

A (−1,−3)B (−1,3)C (0,3)D (0,−3)

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

5

4

9

3

5

A

BCD

Given the equation y= x + 4, what would be the y-value for the corresponding x-value of -5 ?

x y

-5 ?

0 4

5 5

6

0

16

-4

-16

AB

C

D

Given the equation y= -4x - 8, what would be the y-value for the corresponding x-value of 2?

x y

-2 0

0 -8

2 ?

7 The graph of the equation 2x + 6y = 4 passes through point (x,−2). What is the value of x?

A -4B 8C 16D 4

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

8 Point (k,-3) lies on the line whose equation is x - 2y = -2. What is the value of k?

A -8B -6C 6D 8

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

9 Which equation is graphed?

A y = 4x-2B y = -1/2x-4C y = -1/2x+4D y = -2x+4E y = -2x+8

10 Which equation is graphed?

A y = 4x + 1B y = -1/4x + 1C y = 1/4x + 1D y = -4x + 1E y = 4x - 1