Graphing from Tables

A table of values is one of the most useful ways to move from an equation to a graph. A table shows selected input-output pairs, and each row can be written as an ordered pair and plotted on the coordinate plane.

For example, consider:

y = 2x + 1

Choose several values of x and calculate y.

These rows give the points:

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

When the points are plotted, they lie on a straight line.

This process connects three important representations:

equation → table → graph

---

Creating a Table from an Equation

Suppose:

y = 3x + 2

We can choose convenient values for x.

A common choice is:

−2, −1, 0, 1, 2

Now substitute each x-value into the equation.

For x = −2:

y = 3(−2) + 2

y = −6 + 2

y = −4

For x = −1:

y = 3(−1) + 2

y = −3 + 2

y = −1

For x = 0:

y = 3(0) + 2

y = 2

For x = 1:

y = 3(1) + 2

y = 5

For x = 2:

y = 3(2) + 2

y = 8

The completed table is:

---

Turning Table Values into Ordered Pairs

Each row in a table corresponds to an ordered pair:

(x, y)

Using the previous table:

These ordered pairs can be plotted directly on the coordinate plane.

---

Plotting the Points

To plot:

(2, 8)

start at the origin.

Move:

• 2 units right
• 8 units up

Then mark the point.

To plot:

(−2, −4)

move:

• 2 units left
• 4 units down

Remember:

x first, y second

---

Drawing the Linear Graph

If the equation represents a linear relationship, the plotted points should form a straight-line pattern.

After plotting the points:

1. check that they are positioned correctly
2. place a ruler along the points
3. draw a straight line through them
4. extend the line where appropriate
5. add arrows if the relationship continues beyond the displayed graph

The line represents all the points satisfying the equation, not just the points listed in the table.

---

Why We Connect the Points

Suppose the equation is:

y = 2x + 1

The table might contain only:

x = −2, −1, 0, 1, 2

But x could also equal:

0.5

1.7

−3.25

or many other values.

The line represents all of these possible values.

The table gives us enough selected points to construct the graph.

---

Continuous and Discrete Situations

We should not always connect points.

Consider:

y = 4x

If x represents time, x might take any value such as:

1.2 seconds

2.75 seconds

4.5 seconds

The relationship may be continuous, so connecting the points makes sense.

But suppose x represents the number of concert tickets purchased.

You might purchase:

1, 2, 3, or 4 tickets

but not:

2.6 tickets.

This situation is discrete.

The graph may therefore consist of separate points rather than one continuous line.

---

Choosing x-Values

When creating a table, choose x-values that make calculations and graphing convenient.

For:

y = 2x + 3

good choices might be:

−2, −1, 0, 1, 2

These values are:

• small
• easy to calculate
• spread across both sides of zero

Including x = 0 is particularly useful because it reveals the y-intercept.

---

Including Zero

Suppose:

y = 4x − 5

If:

x = 0

then:

y = 4(0) − 5

y = −5

Therefore:

(0, −5)

is on the graph.

This immediately tells us that the y-intercept is:

−5

---

Graphing a Positive Linear Relationship

Consider:

y = 2x + 1

Create a table:

The y-values increase by 2 whenever x increases by 1.

Therefore:

slope = 2

The graph rises from left to right.

---

Recognizing a Pattern in the Table

For:

y = 2x + 1

look at the changes:

Changes in x:

+1, +1, +1, +1

Changes in y:

+2, +2, +2, +2

Therefore:

rate of change = 2/1 = 2

A constant rate of change is a key feature of a linear relationship.

---

Graphing a Negative Linear Relationship

Consider:

y = −3x + 4

Create a table:

As x increases by 1:

y decreases by 3.

Therefore:

slope = −3

The graph falls from left to right.

---

Positive, Negative, and Zero Slope

The pattern in a table can reveal the direction of the graph.

Positive slope

As x increases, y increases.

Example:

y = 3x + 2

Negative slope

As x increases, y decreases.

Example:

y = −2x + 5

Zero slope

As x increases, y remains constant.

Example:

y = 4

---

Graphing a Horizontal Line

Consider:

y = 3

Create a table:

Every point has the same y-coordinate.

The graph is a horizontal line passing through:

y = 3

Its slope is:

0

---

Checking for Linearity in a Table

Consider:

The changes in y are:

+3, +3, +3

while x increases by 1 each time.

The rate is constant.

Therefore, the relationship is linear.

---

A Table That Is Not Linear

Consider:

The changes in y are:

+1, +3, +5, +7

These are not constant.

Therefore, the relationship is not linear.

The plotted points would not lie on one straight line.

---

First Differences

When x-values increase by equal amounts, the changes between consecutive y-values are called first differences.

For a linear relationship, the first differences are constant.

Example:

First differences:

9 − 5 = 4

13 − 9 = 4

17 − 13 = 4

Therefore:

constant first difference = 4

and the relationship is linear.

---

When x Does Not Increase by 1

Suppose:

The y-values increase by 8.

But the x-values increase by 2.

Therefore:

m = Δy/Δx

m = 8/2

m = 4

The slope is 4, not 8.

---

Unequal x-Intervals

Consider:

The changes in y are not equal because the changes in x are not equal.

Calculate the rate:

From the first to second row:

m = (11 − 5)/(3 − 1)

m = 6/2

m = 3

From the second to third:

m = (20 − 11)/(6 − 3)

m = 9/3

m = 3

The rate remains constant.

Therefore, the relationship is linear.

---

Finding an Equation from a Table

Suppose:

The rate is:

4

Therefore:

m = 4

When x = 0:

y = 6

Therefore:

b = 6

The equation is:

y = 4x + 6

This shows how we can move in either direction:

equation → table → graph

or:

graph/table → equation

---

Verifying the Equation

For:

y = 4x + 6

check x = 3:

y = 4(3) + 6

y = 12 + 6

y = 18

The table also gives:

(3, 18)

so the equation agrees with the table.

---

Reading Information from a Graph

Once a graph has been drawn, we can use it to answer questions.

Suppose a graph represents:

d = 50t

where:

d = distance in kilometres

t = time in hours

The point:

(3, 150)

means:

After 3 hours, the object has travelled 150 km.

A graph is not simply a picture. Each point communicates information about the relationship.

---

Reading an Output from a Graph

Suppose you want to find y when:

x = 4

From the x-axis:

1. locate x = 4
2. move vertically to the graph
3. move horizontally toward the y-axis
4. read the corresponding y-value

This gives the output associated with x = 4.

---

Reading an Input from a Graph

Sometimes the output is known instead.

Suppose you want to know:

When is y = 15?

Start with:

y = 15

Then:

1. locate 15 on the y-axis
2. move horizontally to the graph
3. move vertically toward the x-axis
4. read the corresponding x-value

This determines the input that produces y = 15.

---

Interpreting the y-Intercept

Consider:

C = 5h + 20

where C is cost and h is time in hours.

At:

h = 0

C = 20

The graph therefore contains:

(0, 20)

This means there is an initial cost of:

$20

The y-intercept often represents a starting value.

---

Interpreting Slope

For:

C = 5h + 20

the slope is:

5

This means the cost increases by:

$5 per hour

On the graph, every increase of 1 unit horizontally corresponds to an increase of 5 units vertically.

---

Real-World Example: Taxi Cost

Suppose a taxi charges:

• $6 starting fee
• $3 per kilometre

Equation:

C = 3d + 6

Create a table:

Plotting these points produces a straight-line pattern.

The graph shows:

• starting cost = $6
• rate of change = $3/km

---

Interpreting a Point in Context

Using the taxi graph, consider:

(4, 18)

The x-coordinate represents:

4 km

The y-coordinate represents:

$18

Therefore:

A 4 km taxi journey costs $18.

A good interpretation includes both quantities and their units.

---

Real-World Example: Water Tank

A tank initially contains:

300 L

and loses:

20 L each minute

Equation:

V = 300 − 20t

Create a table:

The graph decreases from left to right because the slope is negative.

The point:

(6, 180)

means:

After 6 minutes, 180 L remain in the tank.

---

Realistic Graphing Domains

Mathematically, the equation:

V = 300 − 20t

continues beyond the point where V = 0.

But negative water volume does not make physical sense.

Find when the tank reaches zero:

0 = 300 − 20t

20t = 300

t = 15

Therefore, a realistic graph should stop at:

(15, 0)

when modelling the actual tank.

Context can determine which part of a mathematical graph is meaningful.

---

Real-World Example: Savings

Suppose a student begins with:

$50

and saves:

$15 each week

Equation:

S = 15w + 50

Table:

The graph rises steadily because the student saves at a constant rate.

---

Predicting Beyond the Table

A graph can help estimate values not shown in the original table.

Using:

S = 15w + 50

the table might stop at 5 weeks.

But the line continues.

At 8 weeks:

S = 15(8) + 50

S = 120 + 50

S = $170

This is an example of extrapolation because the prediction is beyond the original table values.

---

Estimating Between Table Values

Suppose a table contains:

If the relationship is continuous and linear, the graph allows us to estimate the value at:

x = 3

The corresponding value is:

y = 30

Estimating between known data values is called interpolation.

---

Graph Scale

Choosing a suitable scale is important.

Suppose the y-values are:

100, 200, 300, 400, 500

Using a scale of 1 unit per grid square would be impractical.

A better scale might be:

50 or 100 units per grid interval

The scale should:

• include all important values
• use the available graph space effectively
• be easy to read
• use consistent intervals

---

Axes Do Not Need the Same Scale

The x-axis and y-axis may use different scales.

For example:

x-axis:

1 square = 1 hour

y-axis:

1 square = 50 km

This is completely acceptable.

However, each individual axis must use a consistent scale.

---

Labelling a Graph

A clear graph should normally include:

• a meaningful title when appropriate
• labelled x-axis
• labelled y-axis
• units
• appropriate scale
• accurately plotted points

For example:

Distance Travelled vs Time

x-axis:

Time

y-axis:

Distance (km)

These labels allow someone to interpret the graph without needing additional explanation.

---

Independent and Dependent Variables

The independent variable is usually placed on the x-axis.

The dependent variable is usually placed on the y-axis.

For example:

If distance depends on time:

x-axis = time

y-axis = distance

If cost depends on number of items:

x-axis = number of items

y-axis = cost

---

A Complete Graphing Process

Suppose:

y = −2x + 5

Choose:

x = −2, −1, 0, 1, 2

Calculate:

Now:

1. draw and label the axes
2. choose an appropriate scale
3. plot (−2, 9)
4. plot (−1, 7)
5. plot (0, 5)
6. plot (1, 3)
7. plot (2, 1)
8. check that the points form a straight-line pattern
9. draw the line through the points

The resulting graph has:

slope = −2

y-intercept = 5

---

Checking Your Graph

After plotting, ask:

Do the points follow the expected pattern?

If one point is far away from the others, check:

• the substitution
• arithmetic
• coordinate order
• signs
• scale
• plotting position

A single misplaced point often indicates a calculation or graphing error.

---

Using the Pattern to Detect Errors

Suppose:

y = 3x + 1

Your table is:

The y-values should increase by 3.

But:

1 → 4 = +3

4 → 7 = +3

7 → 11 = +4

This suggests an error.

Check x = 3:

y = 3(3) + 1

y = 10

Therefore, the final value should be:

10

Pattern recognition provides a useful way to check calculations.

---

Comparing Two Graphs from Tables

Suppose:

Relationship A

Relationship B

Relationship A has:

slope = 3

intercept = 8

Relationship B has:

slope = 5

intercept = 2

A starts higher.

B increases faster.

At x = 3, both have:

y = 17

Their graphs intersect at:

(3, 17)

---

Why Tables Are Useful for Graphing

Tables provide a bridge between symbolic equations and visual graphs.

An equation such as:

y = −4x + 7

is symbolic.

A table converts it into specific points.

A graph then shows the overall relationship visually.

Therefore:

equation → values → coordinates → graph

Each representation gives a different view of the same mathematical relationship.

---

A Reliable Strategy for Graphing from an Equation

1. Write the equation clearly.
2. Create two columns labelled x and y.
3. Choose useful x-values.
4. Substitute each x-value into the equation.
5. Calculate the corresponding y-value.
6. Check for a consistent pattern.
7. Write the rows as ordered pairs.
8. Draw and label the coordinate axes.
9. Choose an appropriate scale.
10. Plot each point accurately.
11. Check whether the points follow the expected pattern.
12. If the relationship is continuous and linear, draw a straight line through the points.
13. Interpret important points in context.

---

Common Mistakes

Reversing coordinates

A table row x = 2, y = 7 produces:

(2, 7)

not:

(7, 2)

Substituting incorrectly

For:

y = 3x − 2

and x = −2:

y = 3(−2) − 2

not:

3(2) − 2

Forgetting parentheses with negative values

Use:

3(−2)

to make the sign clear.

Using an inconsistent scale

Equal distances along an axis must represent equal numerical changes.

Assuming every table is linear

Check the rate of change before drawing a straight line.

Using Δy as the slope without checking Δx

Slope is:

m = Δy/Δx

Connecting points that should be discrete

If x represents whole objects, separate plotted points may be more appropriate.

Drawing a line before checking the points

Check the calculations first.

Ignoring graph labels and units

A graph without labels may be impossible to interpret correctly.

Extending a real-world graph beyond meaningful values

The mathematical equation may continue even when the real situation does not.

---

Key Terms

Table of values — organized list of corresponding input and output values.

Ordered pair — pair of coordinates written as (x, y).

Coordinate plane — two-dimensional system used to plot points.

Plot — place a point at its correct coordinate position.

Linear relationship — relationship with a constant rate of change.

Linear graph — straight-line graph representing a linear relationship.

Slope — rate of change of y relative to x.

y-intercept — value of y when x = 0.

Independent variable — input variable, usually shown on the x-axis.

Dependent variable — output variable, usually shown on the y-axis.

First difference — change between consecutive y-values when x-values are equally spaced.

Scale — numerical value represented by intervals along an axis.

Continuous data — values that can exist anywhere within an interval.

Discrete data — separate countable values.

Interpolation — estimating a value between known data points.

Extrapolation — predicting a value beyond the known data.

---

Key Takeaways

• Tables connect equations to graphs by providing specific coordinate pairs.
• Each row of an x-y table represents an ordered pair (x, y).
• Equations can be used to calculate table values by substitution.
• Including x = 0 often makes the y-intercept easy to identify.
• Linear relationships have a constant rate of change.
• When x increases by equal amounts, linear tables have constant first differences in y.
• If x-intervals are unequal, use Δy/Δx to check the rate.
• A positive slope produces a graph that rises from left to right.
• A negative slope produces a graph that falls from left to right.
• A zero slope produces a horizontal line.
• Points from a linear relationship lie along one straight line.
• Continuous linear relationships are normally represented by connecting the plotted points.
• Discrete situations may require separate points instead of a continuous line.
• Graph axes should be clearly labelled and use appropriate scales.
• The independent variable is usually placed on the x-axis.
• The dependent variable is usually placed on the y-axis.
• Patterns in tables can help detect calculation errors.
• Graphs can be used to estimate values between or beyond table entries.
• Real-world coordinates should be interpreted using the quantities and units shown on the axes.
• A mathematical graph may extend beyond the values that are realistic in a particular context.
• Equations, tables, ordered pairs, and graphs are different representations of the same relationship.

---

Check Your Understanding

1. Create a table for y = 2x + 3 using x = −2, −1, 0, 1, and 2.
2. Write the ordered pairs represented by your table from Question 1.
3. Describe how you would plot the point (−3, 5) on a coordinate plane.
4. Create a table for y = −3x + 4 using x = −2, −1, 0, 1, and 2. Describe the pattern in the y-values.
5. Consider the table:

Determine the rate of change and write an equation representing the relationship.

6. Determine whether this table represents a linear relationship. Explain your reasoning.

7. Consider:

Calculate the slope. Explain why the change in y alone is not the slope.

8. A student creates this table for y = 4x − 1:

Identify the error and correct the table.

9. A taxi's cost is represented by C = 2.5d + 5, where d is distance in kilometres. Create a table for d = 0, 2, 4, 6, and 8 km and explain what the graph would look like.
10. A water tank is represented by V = 240 − 20t, where V is volume in litres and t is time in minutes. Create a table using t = 0, 2, 4, 6, 8, 10, and 12. Describe the pattern and explain what the slope means.
11. Using the water-tank relationship from Question 10, determine when the tank becomes empty. Explain why extending the graph beyond this point would not make sense in the real situation.
12. A graph represents a cyclist's distance from home. The point (3, 45) lies on the graph, where the x-axis shows time in hours and the y-axis shows distance in kilometres. Interpret this point in context.
13. Explain the difference between interpolation and extrapolation. Give an example of each using a table or graph.
14. A relationship contains the points (0, 10), (2, 16), (4, 22), and (6, 28). Determine whether the points form a linear pattern, calculate the slope, find the equation, and predict y when x = 10.
15. A student says, "If I have a table, I can always connect all the points with a straight line." Explain why this statement is incorrect. Discuss both non-linear relationships and the difference between continuous and discrete situations.