C3.2, C3.3; E3.2, E3.3 - Drawing lines and finding gradients

C3.2, C3.3; E3.2, E3.3 - Drawing lines and finding gradients

A straight-line equation describes a whole set of points. You will make tables and draw linear graphs for C3.2 and E3.2, then measure their gradients from a grid for C3.3 and E3.3, including horizontal and vertical lines.

Tables and plotting a straight line

How can a few plotted points reveal a line containing infinitely many points? Try the rule y=2x+1y=2x+1. For every chosen xx, the rule gives exactly one yy. Increasing xx by one increases yy by two, wherever you start. That constant change makes the points line up.

Linear graph

A linear graph is a straight-line graph. A point lies on the graph of an equation when its coordinates make that equation true. For example, (2,5)(2,5) lies on y=2x+1y=2x+1, because 5=2(2)+15=2(2)+1.

Use a table of values to organise the coordinate pairs. Label both axes, choose even scales that fit the values, plot the points carefully and use a ruler to draw one straight line through them. Extend the line over the requested range. Two distinct points fix a straight line; a third provides a useful check for a plotting or arithmetic error. Do not bend a line to include an incorrect point.

[DIAGRAM: asset_name: Tables become straight lines; asset_slug: plotting_lines; caption: Changing the added number moves the line but keeps its slope.; description: Two labelled Cartesian grids show three plotted points joined by a ruled line. The left graph is y equals two x plus one through negative one comma negative one, zero comma one and one comma three. The right is y equals two x minus three through negative one comma negative five, zero comma negative three and one comma negative one.]
Changing the added number moves the line but keeps its slope.

Worked example 1: a table for a rising line

Question. Complete a table for y=2x+1y=2x+1 at x=−1,0,1x=-1,0,1, and describe how to draw its graph.

Plan. Substitute each input, then plot the resulting ordered pairs.

x=−1:y=2(−1)+1=−1bracket the negative inputx=0:y=2(0)+1=1use the same rulex=1:y=2(1)+1=3multiply before adding\begin{aligned} x=-1:&\quad y=2(-1)+1=-1 && \text{bracket the negative input}\\ x=0:&\quad y=2(0)+1=1 && \text{use the same rule}\\ x=1:&\quad y=2(1)+1=3 && \text{multiply before adding} \end{aligned}
xx−1-10011
yy−1-11133

Answer. Plot (−1,−1)(-1,-1), (0,1)(0,1) and (1,3)(1,3), then rule and extend the straight line through them, as in the left grid.

Check. A fresh input x=2x=2 gives y=5y=5. The extended line passes through (2,5)(2,5).

Change only the added constant from +1+1 to −3-3. Every output falls by four; the input values and plotting process stay the same.

Worked example 2: a lower line

Question. Complete a table for y=2x−3y=2x-3 at x=−1,0,1x=-1,0,1, and describe how to draw its graph.

Plan. Recalculate each output using the new constant.

x=−1:y=2(−1)−3=−5substitute carefullyx=0:y=2(0)−3=−3the product is zerox=1:y=2(1)−3=−1subtract three\begin{aligned} x=-1:&\quad y=2(-1)-3=-5 && \text{substitute carefully}\\ x=0:&\quad y=2(0)-3=-3 && \text{the product is zero}\\ x=1:&\quad y=2(1)-3=-1 && \text{subtract three} \end{aligned}
xx−1-10011
yy−5-5−3-3−1-1

Answer. Plot (−1,−5)(-1,-5), (0,−3)(0,-3) and (1,−1)(1,-1); rule and extend the line through them, as in the right grid.

Check. At the additional input x=2x=2, the rule gives y=1y=1. That point is on the drawn line. The outputs are all four below the first table, as expected.

Watch out

The pair (−1,−5)(-1,-5) means x=−1x=-1, y=−5y=-5, not the other way round. Also, join the points with one ruled line rather than stopping after a short segment between two dots.

An equation may be presented in another arrangement when its table is supplied. In that case, use the given coordinate pairs to draw the graph; the meaning of plotting is unchanged.

Gradient as rise divided by run

The gradient measures a line's change in vertical coordinate for each unit increase in horizontal coordinate. On a drawn grid, choose two clear points well apart, then draw a right-angled triangle with a horizontal run and a vertical rise. Read changes in coordinate units, not merely numbers of squares.

Gradient from a grid

For a non-vertical straight line, moving from left to right,

gradient=signed vertical changepositive horizontal change.\text{gradient}=\frac{\text{signed vertical change}}{\text{positive horizontal change}}.

An upward change is positive; a downward change is negative. The horizontal change must be non-zero. Not given in the List of formulas: learn it.

Why can you choose a different-sized triangle? All such right-angled triangles on the same straight line have the same shape: multiplying the run also multiplies the rise by the same factor. Their ratio stays fixed. A long run usually makes a measured gradient more reliable than a tiny triangle.

[DIAGRAM: asset_name: Reading signed rise and run; asset_slug: gradient_triangles; caption: Read each axis scale before dividing the vertical change by the horizontal change.; description: Two numbered grids have horizontal divisions of two coordinate units and vertical divisions of one. Line P rises from negative two comma negative one to four comma two, with a triangle showing run six and rise three. Line Q falls from negative two comma two to four comma negative one, showing run six and rise negative three.]
Read each axis scale before dividing the vertical change by the horizontal change.

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