Coordinate Grids & Graphing
PLOTTING POINTS
Graphing Data
Example 1: Create a line graph based on the following data table:
0 2 4 6 8 10 12
y
Jan
x
Feb
Mar
Apr
May
Step By Step: How to Graph from a Data Table
Step One: Look at the table, what are they trying to tell you with that data? What is its purpose? That is your graph title.
Step Two: Left column of the table is your "x-axis" title and labels. Label your graph accordingly.
Step Three: Right column of the table is you "y-axis" title and labels. Lable your graph accordingly. Make sure the scale (numbers you use on y-axis) is big enough to include all your data values.
Step Four: Begin with finding January on the x-axis and 3° on the y-axis. The point where they meet is where you make a dot on the graph. Repeat for February, March etc.
Step Five: Connect your dots with a line because temperature is a continuous data form. Meaning there are values in-between the dots. (e.g., there are degrees in-between 3° and 5° like 4.5°).
Practice
Practice PDFs available for download:
Plotting Points on a Coordinate Grid
Example 1: Plotting the following coordinates as a shape on a grid
Plot the shape on the grid:
A (1, 2)
B (3, 2)
C (2, 3)
D (3, 3)
​
​
​
​
​
​
1 2 3 4
0 1 2 3 4
x
y
How to plot points A-D on a grid:
Step One: Look at your coordinates for point A (1, 2)
The first number is the x value (1). The second number is the y value (2).
Step Two: Find where 1 is on the x-axis. Then go up to find where 2 is on the y-axis.
Step Three: The point where the meet is point A (1, 2); mark it with a dot.
Step four: Repeat this process for the following points B, C, and D. Connect the points to form the shape.
Practice
Practice PDFs available for download:
TRANSFORMATIONS
Basic Transformations
Example 1: Translate the following shape 2 units right, and 1 unit down:
1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
x
y
x
y
+2 (right)
-1 (Down)
x
y
1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
Step By Step:
Step One: Start with identifying what type of transformation (translation, rotation, or reflection) is required. Here we are using translations (moving points up, down, right & left). We are moving the shape 2 units right, and 1 unit down.
Step Two: Select one point and move it 2 units right (on the x-axis). Then move it 1 unit down (on the y-axis).
Step Three: Repeat this process for all points.
Step Four: Connect the points with straight lines to create the translated shape.
Practice
Practice PDFs available for download:
Successive Transformations
Example 1: Translate the following shape 1 units right, and 2 unit down; twice:
x
y
-2 (Down)
y
1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
x
y
+1 (right)
x
x
y
1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
-2 (Down)
y
+1 (right)
x
x
y
1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
Step By Step:
Step One: Start with identifying what type of successive transformation (translation, rotation, or reflection) is required. Here we are using successive translations (moving points up, down, right & left more than once). We are moving the shape 1 units right, and 2 unit down; twice.
Step Two: Select one point and move it 1 units right (on the x-axis). Then move it 2 unit down (on the y-axis). Do this once more for that point. This would fulfill the successive translation requirement.
Step Three: Repeat this process for the rest of the points.
Step Four: Connect the points with straight lines to create the translated shape.
Practice
Practice PDFs available for download:
LINEAR REALTIONS
Graphing Linear Relation
Example) Graphing a linear equation
The equation of a linear relation is y = -4x + 1
a) Create a table of values for -4 to 4
b) Graph the relation
c) Describe the relationship between the variables in the graph
Step 1) Input values into the equation and solve for y.
When x = -4,
y = -4x + 1
y = -4(-4) + 1
y = 16 + 1
y = 17
When x = -3,
y = -4x + 1
y = -4(-3) + 1
y = 12 + 1
y = 13
When x = -2,
y = -4x + 1
y = -4(-2) + 1
y = 8 + 1
y = 9
Step 2) Fill in the table of values:
... when x is -4, y is 17,
... when x is -3, y is 13,
... when x is -2, y is 9...
Step 3) Graph each ordered pair (x , y) from the table of values:
A(-4, 17)
B(-3, 13)
C(-2, 9) ...
Step 4) Write the relationship between x and y
-
The variables are x and y.
-
When x increases by 1, y decreases by 4.
-
the points lie on a line that goes down and to the right.
-
This is a linear relation because the points form a straight line.
Slope Intercept Form
Slope-Intercept Form of the Equation of a Linear Function
-
y = mx + b
-
m is the slope of the line
-
b is the y-intercept
-
This form is excellent for graphing linear relationships because it provides the slope, and y-intercept point.
Example) Graphing a linear equation given in slope-intercept form
Graph the linear function with the equation: y = x + 3
1
2
Step 1) Compare:
y = x + 3
1
2
With: y = mx + b
m (slope) is
1 Rise
2 Run
Thus, for every 1 point the graph goes up, it will go 2 points right.
b (y-intercept) is 3, with coordinates (0,3)
Step 2) Begin plotting points, starting with the y-intercept (0,3)
So, from (0,3), move 1 unit up and 2 units right, then mark a point.
Draw a line connecting the points.
1
2
Example 2) Writing the Equation of a linear fuction in slope-intercept form given its graph
Write an equation to describe this function.
Verify the equation.
Step 1) Use the equation: y = mx + b.
From the graph solve m (the slope of the line)
From the graph solve b (the y-intercept)
The line intersects the y-axis at -4; so b = -4.
The line also rises -3 points when the run is 2.
So, m =
Substitute for m and b in y = mx + b.
-3
2
y = x - 4
-3
2
Step 2) Verify by plugging in an (x, y) coordinate from the graphed line.
Choose any point such as (- 2, -1).
Substitute:
x = -2
y = -1
-1 = (-2) - 4
-3
2
-1 = 3 - 4
-1 = -1
Since the left side is equal to the right side, the equation is correct
GRAPH ANALYSIS
Domain & Range
Domain: all possible x values in a function
Range: all possible y values in a function
Example 2) Writing the Equation of a linear fuction in slope-intercept form given its graph
Determine the domain and range of this graph.
The dot at each end of the graph indicates that the graph stops at that point.
Step 1) Recall the domain is all x values the function contains. Therefore, look at the graphed line and the x-axis.
Here, the x values consist of -2 to +2, and all real numbers in between.
We write this as:
We say: "x is greater than or equal to -2 and less than or equal to 2."
Step 2) Recall the range is all y values the function contains. Therefore, look at the graphed line and the y-axis.
Here, the y values range from 0 to +2, and all real numbers in between.
We write this as:
We say: "y is greater than or 0 and less than or equal to 2."
QUADRATIC GRAPHS
Vertex Form
Vertex form is a great way to graph a parabola because it tells you important information right away!
The vertex form looks like this:
y = a(x - h) + k
Where:
• (h, k) is the vertex (the highest or lowest point of the parabola).
• "a "tells you how wide or narrow the parabola is and if it opens up ( a > 0 ) or down ( a < 0 ).
2
Example) Graph this equation given in vertex form
For y = 2(x - 3) + 4
2
Step 1) Identify the given information from the vertex form:
h = 3
k = 4
a = 2
​
So, the vertex is at (3, 4)
And the parabola is positive meaning it opens upwards.
if a > 0, the parabola opens up
if a < 0, the parabola opens down
Step 2) Use the 1, 3, 5 method:
This method states, from the vertex, move across 1 unit, then up 1 unit. Next move from that point across 1 unit, then up 3 units. Next from that point across 1 unit, then up 5 units.
​
However, if your "a" value is more than 1, you must multiply 1, 3, 5 by that value. Here our "a" value is 2, so instead of 1, 3, 5, we will move up 2, 6, 10 units every sequence.
​
Lets start here with the vertex (3, 4).
Move across 1 unit to (4, 4). Then up 2 units to A(4, 6).
Move across 1 unit to (5, 6). Then up 6 units to B(5, 12).
Move across 1 unit to (6, 12). then up 10 units to C(6, 22).
1
2
6
10
1
1
(4, 6)
(5, 12)
(6, 22)
Step 4) Find the Axis of symmetry (a vertical line that goes through the vertex) and reflect the points to make the parabola.
Here the axis of symmetry is at the vertex (3, 4)
The point (4, 6) reflected becomes (2, 6)
The point (5, 12) reflected becomes (1, 12)
The point (6, 22) reflected becomes (0, 22)
Axis of symmetry
Step 5) Join the points with a curved line to produce the final parabola.