Linear functions. Gradient
A plane descends at a constant speed. A tank empties at the same rate all the time. In both cases there is a number that sums up "how much the second quantity changes for each unit the first one changes": the gradient. Knowing how to calculate it from any starting data (two points, a table, an equation) is the central tool of this section.
The gradient from two points
Find the gradient of the line passing through the points $A(1, 4)$ and $B(3, 10)$.
Find the gradient of the line passing through the points $A(1, 4)$ and $B(3, 10)$.
Look at the following graph and determine the gradient of the line shown.
Look at the following graph and determine the gradient of the line shown.
The gradient from a table or from a linear function
In a linear function $y = mx + n$, the gradient $m$ is already written in front of the $x$. In a table of values, it is calculated by taking any two rows and applying the same quotient formula.
Given the linear function $y = -3x + 5$, state its gradient and y-intercept. Is it an increasing or a decreasing function?
Given the linear function $y = -3x + 5$, state its gradient and y-intercept. Is it an increasing or a decreasing function?
A linear relationship between two variables is given by the following table of values: | $x$ | $2$ | $4$ | $6$ | |-----|-----|-----|-----| | $y$ | $7$ | $11$| $15$|…
A linear relationship between two variables is given by the following table of values:
| $x$ | $2$ | $4$ | $6$ |
|---|---|---|---|
| $y$ | $7$ | $11$ | $15$ |
Calculate the gradient of this function.
Given the following lines: $y = 2x$, $y = 0,5x$, $y = -x$, $y = -4x$. Order them from least to steepest gradient (absolute value of their gradient) and classify them…
Given the following lines: $y = 2x$, $y = 0,5x$, $y = -x$, $y = -4x$. Order them from least to steepest gradient (absolute value of their gradient) and classify them into increasing and decreasing.
The gradient from the general equation
A line can also be written in general form, $Ax + By + C = 0$. To find its gradient, you rearrange to make $y$ the subject and convert to explicit form $y = mx + n$.
Determine the gradient of the line whose general equation is $4x - 5y + 20 = 0$. Next, write its equation in explicit form.
Determine the gradient of the line whose general equation is $4x - 5y + 20 = 0$. Next, write its equation in explicit form.
Calculate the gradient of the line passing through the points $A(-2,5; 3,2)$ and $B(1,5; -4,8)$.
Calculate the gradient of the line passing through the points $A(-2,5; 3,2)$ and $B(1,5; -4,8)$.
Parallel lines and collinear points
Write the equation of a line parallel to $y = -\frac{3}{4}x + 2$ passing through the point $(-8, 5)$.
Write the equation of a line parallel to $y = -\frac{3}{4}x + 2$ passing through the point $(-8, 5)$.
Check algebraically whether the points $A(-2, 5)$, $B(1, -1)$ and $C(5, -9)$ are collinear by calculating and comparing the gradients of the line segments joining them.
Check algebraically whether the points $A(-2, 5)$, $B(1, -1)$ and $C(5, -9)$ are collinear by calculating and comparing the gradients of the line segments joining them.
The gradient as a rate of change
In a real-life context, the gradient is always interpreted as "how much the quantity $y$ changes for each unit of $x$": speed, consumption, cost per hour... The sign tells you whether it increases or decreases, and the absolute value tells you how quickly it does so.
An aircraft descends at a constant rate to land. $10$ minutes after starting the manoeuvre, it is at an altitude of $8000$ metres, and at $25$ minutes, it is at $3500$…
An aircraft descends at a constant rate to land. $10$ minutes after starting the manoeuvre, it is at an altitude of $8000$ metres, and at $25$ minutes, it is at $3500$ metres. Calculate the gradient of the function relating altitude to time and interpret its meaning in this context.
A spring stretches proportionally to the applied force (Hooke's Law). Hanging a mass of $2$ kg gives a total spring length of $15$ cm. Hanging a mass of $5$ kg gives a…
A spring stretches proportionally to the applied force (Hooke's Law). Hanging a mass of $2$ kg gives a total spring length of $15$ cm. Hanging a mass of $5$ kg gives a total length of $21$ cm. Calculate the gradient of the function relating length to weight, write its general equation, and state the length of the spring at rest (with no weight).