Tables of values
A rule links each input to an output. Keep each pair in the same row or column. Equal input gaps give equal output changes for a linear rule
\[y=a+bx\]
- Substitute Replace the input with the complete value, including its sign \(\displaystyle y=11-2(-3)\)
- Evaluate in order Multiply before adding or subtracting \(\displaystyle \begin{aligned}y&=11-(-6)\\&=17\end{aligned}\)
- Read either way Find one value and read its partner from the same row or column \(\displaystyle (x,y)=(-3,17)\)
- Zero coordinates A pair with zero input lies on the vertical axis. A pair with zero output lies on the horizontal axis \(\displaystyle (0,a)\quad\text{or}\quad(x,0)\)
Worked example
Find the outputs for the two listed inputs and record the pairs
\[\begin{gathered}y=13-4x\\x=-2,\;3\end{gathered}\]
- \(\displaystyle y=13-4(-2)\)Substitute the first input
- \(\displaystyle y=13-(-8)\)Multiply while retaining the fixed term
- \(\displaystyle y=21\)Subtract the negative number
- \(\displaystyle (-2,21)\)Keep this input and output together
- \(\displaystyle y=13-4(3)\)Substitute the second input independently
- \(\displaystyle y=13-12\)Multiply first
- \(\displaystyle y=1\)Subtract
- \(\displaystyle (3,1)\)Record the second pair
Common mistakes
Adding before multiplying
✕\(\displaystyle y=(8+3)(4)=44\)✓\(\displaystyle y=8+3(4)=20\)
Only the coefficient is multiplied by the inputLosing the input sign
✕\(\displaystyle 11-2(-3)=11-6\)✓\(\displaystyle 11-2(-3)=11+6\)
The negative sign belongs to the substituted inputIgnoring the input gap
✕\(\displaystyle \Delta x=2\Rightarrow\Delta y=3\)✓\(\displaystyle \Delta x=2\Rightarrow\Delta y=6\)
For the rule with coefficient three, an input increase of two produces an output increase of sixDifficulty levels
introComplete two or three blanks in short tables with signed integers, including zero
standardComplete two outputs with small exact fractions and read an input from a supplied pair
challengeComplete two outputs, then extend an increasing or decreasing table with non-unit input gaps
examChoose one missing output using small signed integers and exact fractions
Questions about this topic
Must I add the same amount each time?+
Only equal input gaps give equal output changes. Substitution works even when the input gaps differ
How do I read the table backwards?+
Find the output, then read the input in the same row or column
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 4: Linear relations
In this chapter
Construct a table of values from a formulaRead coordinates and points from a straight-line graph4A Points, tables and linear relations: Decide whether a point lies on a lineFind the axis intercepts of a line4C Horizontal, vertical and through-origin lines: Recognise horizontal, vertical and through-origin graphsClassify and compare gradients from graphsFind the gradient of a line through two points4E Direct proportion: Use a direct-proportion model4E Direct proportion: Read and interpret direct-proportion graphsRead the vertical-axis intercept and gradient from an equation4F Gradient-intercept form: Recognise a line or find points for its graphConstruct a line on a coordinate gridFind the equation of a line from its graph or from its intercept and slopeRecover an intercept or recognise a line from a gradient and pointFind the equation of a line through two pointsMidpoint, endpoint and distance of a coordinate segmentForm an equation of a parallel or perpendicular lineDetermine a perpendicular bisector (optional extension)4J Linear modelling: Construct a linear model from a descriptionInterpret the graph of a linear model4K Graphical simultaneous equations: Check a candidate point against two lines (optional extension)Read the point of intersection of two lines (optional extension)