Generate a point on y = 2x − 1 at x = 4
- Substitute x = 4: y = 2(4) − 1.
- Calculate y = 7.
- Plot (4, 7): four horizontally and seven vertically.
Tables, coordinates & graphs
Connect an input-output rule to points on a labelled coordinate plane.
Lesson 06 · Start where you need
Your learning goal
Generate ordered pairs, plot them correctly and choose a meaningful graph for the context.
Name both quantities
A coordinate pair (x, y) places the horizontal coordinate first and the vertical coordinate second. For a bicycle rental, let x be time in hours and y be cost in dollars. Label both axes, include units, and choose a consistent scale. Reversing the coordinates represents a different relationship.
Words worth knowing
- Ordered pair
- An input and its output, written (x, y).
- Rate of change
- Change in output divided by change in input.
- Starting value
- The output when the input is zero.
- Domain
- The set of permitted inputs for the situation.
1
Make the idea visible.
Predict a change, adjust one value, then explain what stayed the same.
The line represents a rule. Use the context to decide which inputs are meaningful.
y = 3x + 8
The same rule in a table
| x | y |
|---|
Build a table from the rule
If y = 3x + 8, inputs 0, 1, 2 and 3 give outputs 8, 11, 14 and 17. Plot (0, 8), (1, 11), (2, 14) and (3, 17). The equation, table and graph are three representations of the same relationship. Check a point by substitution rather than guessing from its position.
Decide whether to connect the points
If any non-negative length of rental time is permitted, joining points can make sense. If x counts complete tickets sold, only whole-number inputs describe actual sales. A straight line may show the underlying rule, but intermediate points need not be possible situations. State the domain you are using.
Read the scale before reading a value
One grid step does not always mean one unit. A graph can have different horizontal and vertical scales. A line that looks steeper after resizing has not changed its numerical rate. Read two coordinate pairs and calculate the changes in the quantities.
Check whether a linear model is justified
For equal input steps, a linear relation has equal output changes. Outputs 3, 7, 11, 15 at inputs 0, 1, 2, 3 have a constant difference of 4. Outputs 1, 4, 9, 16 have differences 3, 5, 7, so one constant-rate line cannot match them all. For unequal input steps, compare change in output divided by change in input. A straight line through two selected points does not establish that every other point follows it.
Worked examples
Follow the reasoning.
Does (3, 18) lie on y = 3x + 8?
- Substitute the horizontal coordinate 3.
- 3(3) + 8 = 17, not 18.
- The point does not lie on this line. The pair (3, 17) does.
Try it with support
Show your thinking.
List three points on y = 2x + 3.
Give me a hint
Choose simple inputs including 0.
Compare a worked response
For example: (0, 3), (1, 5), (2, 7).
Substitute each input to calculate its output.
Should a ticket-sales model include 2.5 tickets? Explain.
Give me a hint
Distinguish a drawn line from permitted inputs.
Compare a worked response
Usually not when counting complete tickets.
The mathematical line extends between points, but this situation uses whole-number ticket counts.
Independent practice
Check your understanding.
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Check the concepts, too.
Choose a response, then use the explanation to check your reasoning. This is practice, not a school mark.
Go further
Keep the connections close.
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