Construct tables and graphs of linear relations

Which ordered pair describes P on this unit grid?

Match each coordinate to its axis

The horizontal axis is x and the vertical axis is y. Their intersection is the origin, with coordinates zero comma zero

Move horizontally to x, then vertically to y. Left and below are negative; right and above are positive

Read the labelled scale rather than assuming one square is always one unit. The grids in these construction tasks use unit spacing

Changing the order generally describes a different point

Which ordered pair describes Q?

Calculate pairs before constructing the line

For each listed x, multiply by two and add one. Keep that input and output together as one ordered pair

Plot at least two distinct correct points and extend a straight line through them in both directions. A third point helps check the calculation and placement

Two copies of the same point do not determine a unique line. The relation y = 2x + 1 is defined for all real x-values, even though the construction controls use integer-coordinate points

Construct a rising line from its equation

Use this linear relation

Complete the three missing y coordinates as signed integers, keeping each supplied x coordinate with its own result

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Construct the straight line defined by the equation. Choose any two distinct integer-coordinate points on the relation and extend the line in both directions. Check your completed table against the line

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Plot at least two distinct points, then draw the straight line through them and check the third point

Other pairs of distinct integer-coordinate points on this same line are valid as well

Construct both the table and a rising line

Check when the line passes through the origin

Use this linear relation

Complete the three missing y coordinates as signed integers, keeping each supplied x coordinate with its own result

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Construct the straight line defined by the equation. Choose any two distinct integer-coordinate points on the relation and extend the line in both directions. Check your completed table against the line

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Plot at least two distinct points, then draw the straight line through them and check the third point

Other pairs of distinct integer-coordinate points on this same line are valid as well

Keep negative inputs and a negative fixed term

Compare output differences at equally spaced inputs

Each input increases by one and each output increases by three. These three points lie on a straight line

For a relation given as linear, equal input changes produce equal output changes. The change may be positive, negative or zero

A finite table alone does not prove that every unlisted value follows the same rule. A stated linear equation justifies extending its straight line

Table

\(\displaystyle \begin{array}{c|rrr}x&-1&0&1\\y&3&1&-1\end{array}\)

Find the change in y when x moves from zero to one

Subtract the earlier output from the later output

Use the actual input spacing

Here the input increases by two, so the output increases by six, not three

The output change per one input unit is three. If input gaps differ, compare output change divided by input change rather than raw output differences

Independent substitution remains valid for any chosen input spacing

Rule

\(\displaystyle y=3x-2\)

Rows

\(\displaystyle (x,y)=(0,-2),\ (2,4)\)

What is the change in y between these two matched rows?

Subtract the output at x zero from the output at x two

Table A

\(\displaystyle \begin{array}{c|rrr}x&-1&0&1\\y&-3&0&3\end{array}\)

Table B

\(\displaystyle \begin{array}{c|rrr}x&-1&0&1\\y&1&0&1\end{array}\)

Which table has all three displayed points on one straight line?

A falling relation still uses the same coordinate order

The coefficient of x is negative one. As x increases by one, y decreases by one

A negative input multiplied by a negative coefficient gives a positive product. Do not drop either minus sign

Plot the calculated pairs in x then y order; a negative coefficient does not reverse the axes

Construct a falling line using signed substitutions

Use this linear relation

Complete the three missing y coordinates as signed integers, keeping each supplied x coordinate with its own result

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Construct the straight line defined by the equation. Choose any two distinct integer-coordinate points on the relation and extend the line in both directions. Check your completed table against the line

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Plot at least two distinct points, then draw the straight line through them and check the third point

Other pairs of distinct integer-coordinate points on this same line are valid as well

Construct a steeper falling relationship

Construct a falling line through the origin

Choose convenient inputs for a fractional coefficient

Using even integer inputs gives integer outputs for this rule, which makes suitable points for the construction grid

The line also contains points with non-integer coordinates. Choosing convenient integer points does not restrict the mathematical relation to those points

Construct a line with a half coefficient

Use this linear relation

Complete the three missing y coordinates as signed integers, keeping each supplied x coordinate with its own result

Use the same equation for each row and retain the signed input

For \(x=-2\)

For \(x=0\)

For \(x=2\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Construct the straight line defined by the equation. Choose any two distinct integer-coordinate points on the relation and extend the line in both directions. Check your completed table against the line

Use the same equation for each row and retain the signed input

For \(x=-2\)

For \(x=0\)

For \(x=2\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Plot at least two distinct points, then draw the straight line through them and check the third point

Other pairs of distinct integer-coordinate points on this same line are valid as well

Use spaced inputs with a three-halves coefficient

Combine a negative half coefficient and a negative fixed term

Use this linear relation

Complete the three missing y coordinates as signed integers, keeping each supplied x coordinate with its own result

Use the same equation for each row and retain the signed input

For \(x=-2\)

For \(x=0\)

For \(x=2\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Construct the straight line defined by the equation. Choose any two distinct integer-coordinate points on the relation and extend the line in both directions. Check your completed table against the line

Use the same equation for each row and retain the signed input

For \(x=-2\)

For \(x=0\)

For \(x=2\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Plot at least two distinct points, then draw the straight line through them and check the third point

Other pairs of distinct integer-coordinate points on this same line are valid as well

Construct a falling line from fractional multiplication

Check a plotted point in the original equation

The calculated output equals the supplied y-coordinate, so P lies on the line

Both coordinates must belong to the same point. Checking only x or only y is not enough

Two distinct points determine a straight line; checking a third point helps detect an incorrectly calculated or plotted pair

Line

\(\displaystyle y=-2x+1\)

Point

\(\displaystyle (-2,y)\)

Find y so that the point lies on the stated line

Substitute negative two for x before calculating y

Relation

\(\displaystyle y=ax+b\)

Change the signed coefficient a and fixed term b. The three marked points use x equal to negative one, zero and one

Why does Q stay on the vertical axis as the settings change?

A horizontal relation fixes y

Every point on this relation has vertical coordinate three. Its horizontal coordinate can vary

Join distinct points at the same height and extend a horizontal line. The constant output has zero change for every input step

The relation y = 3 is not x = 3; the letter tells you which coordinate is fixed

Construct a horizontal line above the origin

Use this linear relation

Complete the three missing y coordinates as signed integers, keeping each supplied x coordinate with its own result

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Construct the straight line defined by the equation. Choose any two distinct integer-coordinate points on the relation and extend the line in both directions. Check your completed table against the line

Use the same equation for each row and retain the signed input

For \(x=-1\)

For \(x=0\)

For \(x=1\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Plot at least two distinct points, then draw the straight line through them and check the third point

Other pairs of distinct integer-coordinate points on this same line are valid as well

Construct a horizontal line with a negative fixed coordinate

A vertical relation fixes x

Every point has horizontal coordinate negative three, while its vertical coordinate can vary

The table now supplies y values and asks for x. The plotted coordinate order is still x then y

A vertical line is a linear relation, but it does not assign a single y to each x. Do not try to write it as y = ax + b with a finite coefficient a

Construct a vertical line left of the origin

Use this linear relation

Complete the three missing x coordinates as signed integers, keeping each supplied y coordinate with its own result

Use the same equation for each row and retain the signed input

For \(y=-2\)

For \(y=0\)

For \(y=2\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Construct the straight line defined by the equation. Choose any two distinct integer-coordinate points on the relation and extend the line in both directions. Check your completed table against the line

Use the same equation for each row and retain the signed input

For \(y=-2\)

For \(y=0\)

For \(y=2\)

Keep the coordinates in x then y order when plotting, even when the supplied input column was y

Plot at least two distinct points, then draw the straight line through them and check the third point

Other pairs of distinct integer-coordinate points on this same line are valid as well

Keep x fixed while choosing different y values

Relation

\(\displaystyle x=2\)

Which statement is true for every point on this relation?

The coordinate axes are special constant relations

A point on the horizontal axis has no vertical displacement. A point on the vertical axis has no horizontal displacement

The axis name refers to the coordinate that varies along it, not the coordinate fixed at zero

The origin belongs to both axes, but one point by itself does not identify which whole axis is intended

Construct the horizontal coordinate axis

Construct the vertical coordinate axis

Explain a complete graph construction

Independent written task, not automatically marked

Choose a signed linear rule with a numerical coefficient and fixed term. State that the relation is defined for all real x values

Calculate at least three matched coordinate pairs, including a negative input. Label and scale both axes, plot the points and extend the straight line

Use substitution to check each pair, and explain what changes when the input step is not one

Construct a horizontal relation and a vertical relation separately, naming the fixed coordinate in each

Explain why two distinct points determine the straight line but a single point does not. State why sampled values alone do not determine an unstated rule outside the table

Review the drawings and explanation against these instructions or with a teacher; the automatic construction checks assess the supplied relations

Keep equation, table and graph consistent

Substitute each signed input into the stated rule and keep its coordinates together in x then y order

Use two distinct correct points to construct the line, and check another point by substitution

For equal input steps, a given linear relation has equal output differences. Unequal input gaps require comparing changes per input unit

Convenient inputs can produce integer-coordinate points even when a coefficient is fractional

Horizontal relations fix y; vertical relations fix x. On the horizontal axis, y is zero; on the vertical axis, x is zero

Extend a line because the stated relation is linear, not because a finite table alone proves an unstated rule