Algebra Review - Graphs and Linear Functions

1. Plotting Points

The perpendicular lines are axes. The horizontal is the x-axis and the vertical the y-axis.

The origin is (0,0). The coordinate plane is divided in four quadrants and it is important that you remember the correct order!

To read a set of coordinates, you state how much you move horizontally and then vertically. If you move left of the origin, the number is negative. Moving right is positive. Moving up is positive and down is negative.

2. Plotting Points from an Equation

Basically just make a table and plug in at least two numbers. Use more to get a more accurate line if you are drawing the graph by hand.

The equation: y = 2x +1

Make a table:




2 5 (2,5)
1 3 (1,3)
0 1 (0,1)
-1 -1 (-1,-1)
-2 -3 (-2,-3)

Then plot the points.

Equation: y = 3x - 1





2 5 (2,5)
1 2 (1,2)
0 -1 (0,-1)
-1 -4 (-1,-4)
-2 -7 (-2,-7)


3. Intercepts in Plotting

Intercepts are the points where the line crosses the axes. The y-intercept is where the line crosses the y-axis and the x-intercept is where it crosses the x-axis. Using only these two points it is possible to successfully plot a graph without plugging in many numbers.

Equation: y = x - 3





0 -3 (0,-3)
3 0 (3,0)

Then just plot.

Equation: y = -2x + 1
   Find the intercepts:

y = -2(0) + 1
   = 1
0 = -2x + 1
2x = 1
x = .5





0 1 (0,1)
.5 0 (.5,0)

4. Finding the Slope of a Line from Two Points

The slope is the slant of the line and in definition is:

Take the following two points: (3,5) and (5,9).
Use the slope equation: . You find out the slope is two. By definition it means that for every unit of x that increases, y is increased by two units.

4. Finding the Slope of a Line from an Equation

To find the slope of an equation, such as 3x + 2y = 1 and x - 4y = -2, solve for the y variable and the coefficient of the x variable is the slope.

The slope is -3/2.
The slope is 1/4.
You should notice that when the slope is positive, the line slants to the upper right. Whereas when the slope is negative, the line slants to the upper left.

4. Finding the Equation

The basic slope equation is: y = mx + b where m is the slope and b is the y-intercept.

Write an equation for a line that has slope 2 and passes through (1,2).

Use the basic equation.

Plug in for the slope (2).

Plug in the point (1,2) in the right variables.

Simplify and solve for b.

You find the answer for b is 0.

When you are given two points, you can still find the equation of the line.

Find the equation from the points (1,4) and (-5,-2).

Find the slope:

Then plug in. You can use either set of points:

Your final answer is y = x + 3

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