Tables of Values & Variable Relationships
Match Rule to Table Output
Draw a line from each rule to the correct output value when x = 3.
Complete the Table
Find the y value for each rule.
y = 2x + 1, x = 4
y = 3x − 2, x = 5
y = x² + 1, x = 3
y = x ÷ 2 + 3, x = 8
Find the Pattern in the Table
Fill in the missing y values using the rule y = x + 4.
Which Rule Matches?
A table has x: 1,2,3,4 and y: 3,6,9,12. Sort: which rules fit?
Increasing or Decreasing?
As x increases, does y increase, decrease, or stay the same?
y = 2x
y = 10 − x
y = x²
y = 5
Create Your Own Table of Values
Complete each table of values and describe the pattern.
Rule: y = 4x − 1. Complete the table for x = 0, 1, 2, 3, 4. Describe what happens to y as x increases.
A taxi charges $3 plus $2 per km. Write a rule for total cost C in terms of km k. Fill in a table for k = 1, 2, 3, 4, 5.
Find the Rule from the Table
Match each table of values to its rule.
Non-Linear Patterns (x²)
For the rule y = x², circle the correct output.
y = x², x = 5
y = x², x = 7
y = x² − 1, x = 4
y = 2x², x = 3
Match Table to Graph Description
Sort each rule to the graph type it would produce.
Describing Rates of Change
Analyse how variables change.
A rule has y values: 1, 4, 9, 16, 25 for x = 1, 2, 3, 4, 5. Is the rate of increase constant? Explain what this tells you about the rule.
Two mobile plans: Plan A costs $10 + $0.20/min. Plan B costs $0.50/min. Build a table for 0, 10, 20, 30 min. Which is cheaper for 20 min? 30 min?
Rate of Change
Circle the correct rate of change.
y = 3x + 1: for every 1 unit increase in x, y increases by:
y = x − 5: for every 1 unit increase in x, y:
y = 7: as x increases, y:
y = −2x + 8: as x increases by 1, y:
Plotting Tables of Values
Complete each table and plot the points on a coordinate grid.
Rule: y = 2x − 1. Complete for x = −2, −1, 0, 1, 2, 3. Plot on a coordinate grid and draw the line.
Rule: y = x² for x = −3, −2, −1, 0, 1, 2, 3. Plot on a coordinate grid. Describe the shape.
Gradient and y-intercept
Identify the gradient and y-intercept of each rule, then describe the line.
y = 3x − 4: gradient = ___, y-intercept = ___. Describe the line's steepness and direction.
y = −2x + 6: gradient = ___, y-intercept = ___. Describe the line.
y = ½x: gradient = ___, y-intercept = ___. How does this line compare to y = 2x?
Steeper or Shallower?
Sort these rules from the least steep to the steepest positive gradient.
Real-World Tables — Distance and Speed
Build and analyse a table of values for this real-world situation.
A car travels at 90 km/h. Complete a table for t = 0, 1, 2, 3, 4 hours: time (t) → distance (d). Write the rule d = ___.
A second car starts 50 km ahead and travels at 60 km/h. Write its rule and add its values to the same table. When does the first car catch the second?
Cost Comparison Using Tables
Build tables and compare two plans.
Plan A: $25 per month plus $0.10 per text. Plan B: $15 per month plus $0.20 per text. Build tables for 0, 50, 100, 150, 200 texts. At what number of texts do the plans cost the same?
Write the rule for each plan. Solve the equation Plan A = Plan B algebraically and verify with your table.
Quadratic Tables
Build and analyse a table for a quadratic rule.
Rule: y = x² − 2x + 1. Complete for x = −2, −1, 0, 1, 2, 3, 4. What pattern do you see in the second differences of y?
For what value(s) of x is y = 0? What does this tell you about the graph?
Finding Intersection from Two Rules
Find the point where two rules give the same y value.
y = 2x + 3 and y = 5x − 6. Build tables for x = 0 to 5. Where do the rules give the same value? Solve algebraically to verify.
Two factories: Factory A produces 200 + 50n items per day (n = days), Factory B produces 150n items. When do they produce the same amount?
Non-Linear Tables — Second Differences
Use second differences to identify the type of rule.
For y = x²: x = 1,2,3,4,5. Find first differences (Δy) and second differences (Δ²y). What do you notice?
Given table: x: 1,2,3,4,5 → y: 3,7,13,21,31. Find first and second differences. What type of rule does this suggest?
Composite Function Machines
Apply two rules in sequence.
Rule 1: multiply by 2. Rule 2: add 5. Complete: input 3 → after Rule 1 → after Rule 2 → output. Do this for inputs 1, 2, 3, 4, 5. Write the combined rule.
If you reverse the order (add 5 first, then multiply by 2), do you get the same result? Investigate with x = 3.
Design a Data Table Investigation
Create your own investigation using a table of values.
Choose a real-world situation (e.g., cost of buying items, distance walked over time). Design a table with at least 6 rows. Collect or estimate the data.
Find the rule (or formula) that best describes your data. Is it linear or non-linear?
Use your rule to predict a value beyond your table. Check if the prediction is reasonable.
Domain and Range in Context
Analyse the domain and range of each real-world rule.
A bathtub fills at 12 L/minute. Rule: V = 12t where V is volume and t is time. What is a reasonable domain? What is the range if the tub holds 240 L?
A ball is thrown up: h = −5t² + 20t (height h in metres, time t in seconds). What is the realistic domain (when is the ball in the air)? What is the maximum height?
Non-Linear Functions — Quadratics
Investigate the quadratic rule y = ax² + bx + c.
For y = x² − 4, find: where it crosses the x-axis (set y = 0), the y-intercept (set x = 0), and the minimum value. Build a table for x = −3 to 3 and sketch the graph.
How does changing a in y = ax² affect the shape? Compare y = x², y = 2x², y = ½x² using a table.
Project: Predicting with Rules
Use mathematical rules to make and test predictions.
Record or estimate the cost of electricity (in kWh) for 5 different household uses. Build a table (usage → cost). Find the rule. Predict the monthly cost for a typical household.
Evaluate: how well does your linear rule fit the data? Are there non-linear effects? Discuss.
Reflection: Tables of Values
Summarise your understanding.
How do you determine the rule from a table of values? Describe the process step by step.
What is the difference between a linear and a non-linear rule? Give an example of each.
Describe a real situation where building a table of values helped you solve a problem or make a decision.
Real-World Tables and Rules
Find examples of input-output relationships at home.
- 1Look at a mobile phone plan. Create a table showing cost for 0, 5, 10, 15, 20 GB of data. Write the rule.
- 2Record the temperature at the same time each day for 5 days. Is the pattern linear? Why or why not?
- 3Time how long it takes to walk 100 m. Create a table for 100 m, 200 m, 300 m, 400 m. What is the rule?
- 4Find a price list (e.g., fruit per kg) and write a rule for cost of x kg. Build a 5-row table.
- 5Research fuel efficiency for a car: build a table of (distance driven) → (fuel used). Is the relationship linear?
Extending — Interpreting Gradient and Intercept
Explain what the gradient and y-intercept mean in each real-world context.
A taxi charges a $4 flag fall plus $2.50 per km. The rule is C = 2.5d + 4. What does the gradient mean? What does the y-intercept mean?
A swimmer trains and improves by 0.3 seconds per week. If their initial time is 52 seconds, write the rule T = a − 0.3w. What does the negative gradient mean in context?
Write your own real-world linear rule with gradient −5. Explain what the negative gradient means.
Extending — Comparing Linear and Non-Linear Growth
Use tables to compare two rules side by side.
Build tables for x = 0, 1, 2, 3, 4, 5 for both y = 3x + 1 and y = x² + 1. Display them side by side.
For each table, find the first differences (change in y as x increases by 1). How do they differ?
At what x value does the quadratic rule overtake the linear rule?
Extending — Reverse Engineering Tables
Find the rule, then use it to predict a distant term.
x: 1, 2, 3, 4 | y: 3, 7, 13, 21. Find the second differences. Write the quadratic rule y = ax² + bx + c.
Use your rule to predict the output when x = 10.
Check your rule by substituting x = 1, 2, 3, 4 and confirming the outputs match.
Extending — Identifying the Steepest Rule
Circle the rule with the steepest gradient (fastest rate of change) for x between 0 and 5.
Which grows fastest initially?
Which rule would give the largest y when x = 100?
A gradient of −4 means the output:
Extending — Two-Variable Relationships
Explore relationships involving two input variables.
The rule z = 2x + 3y. Complete a table for x = 1, 2, 3 and y = 1, 2. (That is 6 combinations — lay it out as a grid.)
Describe how z changes as x increases by 1 (with y constant). Then describe how z changes as y increases by 1 (with x constant).
Extending — Sort Tables by Type of Relationship
Sort each described table into the correct category.
Extending — Tables and Simultaneous Equations
Use tables to solve a pair of simultaneous linear equations graphically.
Build tables for y = x + 2 and y = 3 − x, for x = 0, 1, 2, 3. Find where the outputs are equal.
What are the x and y values at the intersection? Write the solution as an ordered pair (x, y).
Verify algebraically: solve x + 2 = 3 − x and check the solution matches the table.
Extending — Design a Rate Investigation
Design and carry out your own investigation into a real-world rate.
Choose a real-world rate (e.g., water filling a container, steps walked per minute, words typed per second). Measure 5 data points and record them in a table.
Plot your data points. Is the relationship roughly linear? Find the gradient (rate) and write the rule.
Use your rule to make a prediction for a value outside your measurements. Discuss how confident you are in this prediction.
Extending — Piecewise Rules
Some real-world relationships have different rules for different input ranges.
A taxi charges $3 for the first km and $1.50 for each additional km. Write two rules: one for d ≤ 1 and one for d > 1. Build a table for d = 1, 2, 3, 4, 5.
Sketch a graph of the fare against distance. What does the 'kink' in the graph represent?
Extending — Mini-Project: Predict the Future
Use a table of values to make a prediction and evaluate its reliability.
Find real data online for a quantity that changes over time (e.g., Australia's population by decade, smartphone sales by year, CO₂ levels by year). Record 5 data points in a table.
Plot the data and decide whether a linear or non-linear rule is more appropriate. Write your rule.
Use your rule to predict the value 10 years from now. Discuss one reason your prediction might be wrong.