FA9–FA10, HA9–HA10 - Gradient and intercept

FA9–FA10, HA9–HA10 - Gradient and intercept

Can two lines cross the vertical axis at different heights yet stay the same distance apart? FA9 and HA9 connect straight-line equations to parallel lines. FA10 and HA10 explain how gradient and intercept describe a line's direction and starting value.

Gradient as a ratio

A line that rises by three units whenever you move two units right has gradient 3/23/2. Moving twice as far gives rise six and run four: the ratio is unchanged.

Gradient

The gradient of a non-vertical straight line is its change in yy divided by its change in xx:

m=change in ychange in x=y2−y1x2−x1.m=\frac{\text{change in }y}{\text{change in }x}=\frac{y_2-y_1}{x_2-x_1}.

Use the same point order in both differences. The denominator must not be zero.

Read axis values, rather than counting squares without checking their scales. Choose two well-separated points exactly on the line and draw a horizontal-and-vertical gradient triangle. Similar gradient triangles have the same rise-to-run ratio, which is why any pair of points on a straight line gives the same gradient.

[DIAGRAM: asset_name: Positive and negative gradient triangles; asset_slug: gradient-triangles; caption: Gradient compares coordinate changes, including a negative change in height.; description: Two scaled coordinate grids show a rising line through (1,2) and (5,8) with run 4 and rise 6, and a falling line through (-2,7) and (4,3) with run 6 and change in y minus 4.]
Gradient compares coordinate changes, including a negative change in height.

Worked example 1: a rising line

Question. A straight line passes through (1,2)(1,2) and (5,8)(5,8). Calculate its gradient.

Plan. Take the changes from the first point to the second.

m=8−25−1=64=32.m=\frac{8-2}{5-1}=\frac64=\frac32.

The positive numerator and denominator describe a line rising as you move right.

Answer. m=3/2m=3/2, or 1.51.5.

Check. Reversing both orders gives (2−8)/(1−5)=(−6)/(−4)=3/2(2-8)/(1-5)=(-6)/(-4)=3/2, the same ratio.

Now change the direction of the line. The vertical change must keep its sign.

Worked example 2: a falling line

Question. Find the gradient through (−2,7)(-2,7) and (4,3)(4,3).

Plan. Move right from x=−2x=-2 to x=4x=4, recording the fall in yy.

m=3−74−(−2)=−46=−23.m=\frac{3-7}{4-(-2)}=\frac{-4}{6}=-\frac23.

Answer. −2/3-2/3. For every three units right, the line falls two units.

Check. Moving six right should give change 6(−2/3)=−46(-2/3)=-4, taking 77 down to 33.

A horizontal line has zero vertical change, so its gradient is zero. A vertical line has zero horizontal change; its gradient is undefined, because division by zero is undefined. It does not have gradient zero.

Watch out

Using (7−3)/(4−(−2))(7-3)/(4-(-2)) in the second example reverses only the numerator and wrongly gives a positive gradient. Subtract coordinates in the same order on the top and bottom.

Try another falling line.

The y-intercept

The gradient describes how a line changes; its intercept fixes where the line sits.

Slope-intercept form

In y=mx+cy=mx+c, mm is the gradient and cc is the y-intercept, the value of yy where the line crosses the vertical axis. The crossing point is (0,c)(0,c), because every point on the y-axis has x=0x=0.

For example, y=2x+3y=2x+3 starts at (0,3)(0,3). Increasing xx by one increases yy by two. Plot the intercept and use the gradient to locate another point, then join with a ruler. A third substitution checks that the line is correct.

[DIAGRAM: asset_name: Reading a line equation; asset_slug: line-intercept; caption: The intercept locates the line, and the gradient gives its change per unit across.; description: A scaled graph of y equals minus one half x plus 4 marks (0,4), (2,3) and (8,0), with a gradient triangle of run 2 and fall 1. The vertical and horizontal intercepts are distinguished.]
The intercept locates the line, and the gradient gives its change per unit across.

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