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E3.3 Gradient of linear graphs

Syllabus
0580–2028–2029
Topic
E3.3
Level
Extended

Read a line’s gradient as rise over run

The gradient measures how much yy changes for each unit increase in xx. Read the line from left to right: a rising line has positive gradient and a falling line has negative gradient.

gradient=vertical changehorizontal change=riserun\text{gradient}=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{\text{rise}}{\text{run}}

Choose two clear grid points on the line. From the left point to the right point, record the horizontal change first and the signed vertical change second. Divide vertical change by horizontal change using the actual axis values, not merely the number of grid squares.

If moving 44 units right along a line requires moving 33 units down, the horizontal change is +4+4 and the vertical change is −3-3. The gradient is therefore −3/4=−0.75-3/4=-0.75.

A horizontal line has gradient 00 because its vertical change is 00. A vertical line has no defined gradient because its horizontal change is 00, which would require division by zero.

Do not reverse the fraction or discard the sign. Any two exact points on the same straight line give the same gradient, but estimated or misread points can change the result.

Calculate gradient from two coordinates

For two points on a straight line, calculate gradient by dividing their change in yy by their change in xx. Both differences must use the same point order.

m=y2−y1x2−x1for (x1,y1) and (x2,y2)m=\frac{y_2-y_1}{x_2-x_1}\qquad\text{for }(x_1,y_1)\text{ and }(x_2,y_2)

Label the two coordinates, subtract the yy-values for the numerator, then subtract the corresponding xx-values in the same order for the denominator. Keep brackets around negative coordinates before simplifying.

(−2,−7), (4,1):m=1−(−7)4−(−2)=86=43(-2,-7),\ (4,1):\qquad m=\frac{1-(-7)}{4-(-2)}=\frac{8}{6}=\frac{4}{3}

Reversing both subtractions gives (−7−1)/(−2−4)=(−8)/(−6)=4/3(-7-1)/(-2-4)=(-8)/(-6)=4/3, so the gradient is unchanged. Reversing only one subtraction incorrectly changes the sign.

If the two xx-coordinates are equal, the denominator is zero and the line is vertical, so its gradient is undefined. Otherwise give the fraction in its simplest form unless a decimal is required.