Linear and affine functions
A taxi in Cork charges €3.50 for getting in and €1.20 per kilometre. A gym charges a €20 joining fee and €15 a month. Your savings for the Transition Year trip grow by €5 every week. In all these cases the graph is a straight line, and with two numbers (how much it rises per unit and where it starts) you know everything.
Gradient and y-intercept
Move $m$ and $c$ (the slider labelled $n$). Notice that $m$ changes the steepness and $c$ raises or lowers the line without turning it:
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$m > 0$: increasing. $\quad m < 0$: decreasing. $\quad m = 0$: horizontal line (constant function).
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The greater $|m|$ is, the steeper the line.
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Two lines with the same slope are parallel. Two lines are perpendicular if their slopes multiply to give $-1$, for example $2$ and $-\frac{1}{2}$.
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A constant function such as $y = 4$ has slope $0$: it is a horizontal line, so it does have a slope, and that slope is $0$. A line with $m < 0$ cannot pass through the origin unless $c = 0$, so having a negative slope says nothing about the origin.
For each of the following functions, state the value of the gradient and the y-intercept, and indicate whether its graph is increasing or decreasing: a) $y = 4x - 3$ b)…
For each of the following functions, state the value of the gradient and the y-intercept, and indicate whether its graph is increasing or decreasing:
- a) $y = 4x - 3$
- b) $y = -2x + 7$
- c) $y = \frac{1}{3}x$
- d) $y = 5 - x$
Of the following equations, state which correspond to a linear function, which to an affine function, and which are neither linear nor affine. Justify your answer. a)…
Of the following equations, state which correspond to a linear function, which to an affine function, and which are neither linear nor affine. Justify your answer.
- a) $y = -3x$
- b) $y = x^2 + 1$
- c) $y = \frac{x}{2} - 4$
- d) $y = \frac{2}{x}$
- e) $y = 7$
The figure shows three lines, $a$, $b$ and $c$, as shown below. a) Match each line with its equation from these three: $y = 2x + 1$, $y = -x + 3$,…
The figure shows three lines, $a$, $b$ and $c$, as shown below.
- a) Match each line with its equation from these three: $y = 2x + 1$, $y = -x + 3$, $y = \frac{1}{2}x - 2$.
- b) Explain how you used the sign of the gradient and the y-intercept to decide.
Calculating the slope
If you know two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the line:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$$Calculate the gradient of the line passing through each pair of points: a) $A(2, 5)$ and $B(6, 13)$ b) $P(-1, 4)$ and $Q(3, -8)$ c) $M(0, -2)$ and $N(5, -2)$
Calculate the gradient of the line passing through each pair of points:
- a) $A(2, 5)$ and $B(6, 13)$
- b) $P(-1, 4)$ and $Q(3, -8)$
- c) $M(0, -2)$ and $N(5, -2)$
Find the equation of the line passing through the points $A(1, 2)$ and $B(3, 8)$ by following these steps: a) Calculate the gradient $m$. b) Substitute the coordinates…
Find the equation of the line passing through the points $A(1, 2)$ and $B(3, 8)$ by following these steps:
- a) Calculate the gradient $m$.
- b) Substitute the coordinates of one of the points into $y = mx + n$ to find $n$.
- c) Write the final equation and check that the other point also satisfies it.
a) Find the equation of the line passing through the points $A(-1, 7)$ and $B(2, -5)$. b) Check whether the point $C(5, -17)$ is collinear with $A$ and $B$. c) What…
- a) Find the equation of the line passing through the points $A(-1, 7)$ and $B(2, -5)$.
- b) Check whether the point $C(5, -17)$ is collinear with $A$ and $B$.
- c) What about the point $D(0, 5)$? Justify your answer without plotting.
Drawing a line
Two points are enough. Make a table with two simple values of $x$ (for example, 0 and 1, or 0 and another that gives whole numbers), calculate their $y$ values, plot the points and join them.
To check whether a point is on the line, substitute: $(1, 4)$ is on $y = -2x + 6$ because $-2 + 6 = 4$ ✓.
Check, by substituting the coordinates into the equation, whether the points $A(3, 4)$ and $B(-2, 11)$ lie on the line $y = -3x + 5$.
Check, by substituting the coordinates into the equation, whether the points $A(3, 4)$ and $B(-2, 11)$ lie on the line $y = -3x + 5$.
Copy and complete the table of values of the function $y = -2x + 6$ in your notebook for $x = -1$, $0$, $1$, $2$ and $3$, and plot the line on Cartesian axes. At what…
Copy and complete the table of values of the function $y = -2x + 6$ in your notebook for $x = -1$, $0$, $1$, $2$ and $3$, and plot the line on Cartesian axes. At what point does it cross the $X$ axis?
The graph shows an affine function, as shown in the figure. a) What is its y-intercept? b) By how much does $y$ increase when $x$ increases by one unit? What is the…
The graph shows an affine function, as shown in the figure.
- a) What is its y-intercept?
- b) By how much does $y$ increase when $x$ increases by one unit? What is the gradient?
- c) Write its equation.
The line in the figure represents a linear function, as shown below. a) Find two points with integer coordinates that the line passes through. b) Calculate its gradient…
The line in the figure represents a linear function, as shown below.
- a) Find two points with integer coordinates that the line passes through.
- b) Calculate its gradient using those points.
- c) Write the equation of the function.
Lines in real life
In problems, $c$ is usually the fixed value (the joining fee, the starting charge) and $m$ the value per unit (euros per month, euros per km), also called the rate of change. A speed is a rate of change too: $24$ km/h means $m = 24$ in $y = 24x$ (distance against time).
A gym charges a registration fee of $\text{€}\ 20$ and $\text{€}\ 15$ per month. a) Write the equation giving the total cost $y$ in euros as a function of the number of…
A gym charges a registration fee of $\text{€}\ 20$ and $\text{€}\ 15$ per month.
- a) Write the equation giving the total cost $y$ in euros as a function of the number of months $x$.
- b) What is the y-intercept and what does it represent in this context?
- c) What about the gradient? What does it mean?
A taxi charges a flat fee of $\text{€}\ 3.50$ and $\text{€}\ 1.20$ per kilometre travelled. a) Write the function giving the price $y$ for a journey of $x$ kilometres.…
A taxi charges a flat fee of $\text{€}\ 3.50$ and $\text{€}\ 1.20$ per kilometre travelled.
- a) Write the function giving the price $y$ for a journey of $x$ kilometres.
- b) How much will a journey of $12\ \text{km}$ cost?
- c) If a journey cost $\text{€}\ 16.70$, how many kilometres were travelled?
Find the point of intersection of the lines $y = 2x + 1$ and $y = -x + 7$: a) Set up the equation that results from equating the two expressions and solve it. b)…
Find the point of intersection of the lines $y = 2x + 1$ and $y = -x + 7$:
- a) Set up the equation that results from equating the two expressions and solve it.
- b) Calculate the y-coordinate of the intersection point.
- c) Check the result by substituting into both equations.
Back to the start: The taxi follows the line $y = 1.20x + 3.50$ (€ according to km) and the gym follows $y = 15x + 20$ (€ according to months); in both cases, the fixed fee is the y-intercept ($c$) and the cost per unit is the slope ($m$).