Quick questions on Linear functions and the straight line for HSC Maths Advanced: gradient as rise over run and as tan of the angle of inclination, the gradient-intercept, point-gradient and general forms, parallel (equal gradients) and perpendicular (product of gradients minus one) conditions, and the length (distance) and midpoint of an interval, with worked examples and practice questions

4short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is gradient?

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The gradient of an interval (or of the line through it) measures its steepness as you walk from one point to the other. It is the rise (the vertical change) divided by the run (the horizontal change). For two points

P(x_1, y_1)

and

Q(x_2, y_2)

What is gradient as the tangent of the angle of inclination?

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There is a second, equivalent way to read a gradient. The angle of inclination

\theta

is the angle the line makes with the positive

x

-axis, measured anticlockwise. Because the gradient triangle is a right-angled triangle with the rise opposite

\theta

and the run adjacent to it,

What is the three forms of the equation of a line?

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NESA expects you to move fluently between three standard forms. Each is best for a different job.

What are lines in real contexts?

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A straight line is the natural model whenever a quantity changes at a constant rate. Suppose a Sydney plumber charges a fixed callout fee of $90 plus $75 for each hour on site. The total cost

C

for

h

hours is

C = 90 + 75h

. This is a line in disguise: the gradient

75

is the rate (the cost per extra hour) and the intercept

90

is the starting value (the cost before any hours are billed).

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