Mathematics · Functions and graphs · Grade 7

The linear function

Taxi fare: a starting charge + a rate per km. Why is its graph a straight line?

The situation

A taxi meter: 5 000 soum when you get in, 2 000 soum for every kilometre — how much do you pay for the distance you travel? A reservoir holds 300 litres, and 20 litres flow out every minute — when will it be empty? In both questions one number depends on another, and the relationship is drawn as a straight line. Why exactly a straight line, and what do its slope and starting point mean?

Topic

The Cartesian coordinate system, the concept of a function, the linear function and its graph. A taxi meter and a reservoir: starting fare + price per km — the straight line y = kx + b, the gradient k, the starting value b

Key ideas

Coordinates
Function
y = kx + b

Formulas

A (x; y)

Coordinates of a point: the abscissa $x$ (horizontal axis) and the ordinate $y$ (vertical axis); the axes divide the plane into four quadrants.

y = f (x)

Function: to each value of $x$ (the argument) there corresponds exactly one $y$ (the value of the function).

y = k x

Direct proportion: its graph is a straight line through the origin; $k$ is the gradient.

y = k x + b

Linear function: the graph of $y = k x$ raised by $b$ is a straight line; $(0; b)$ is its point of intersection with the $O y$ axis.

k = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}, b = y_{1} - k x_{1}

The formula from two points: $k$ is the change in $y$ relative to the change in $x$ , then $b$ .

Theory

Cartesian coordinate system

Two perpendicular number lines — the abscissa axis $O x$ and the ordinate axis $O y$ — meet at the point $O$ , the origin. Every point of the plane corresponds to a pair of numbers $(x; y)$ : like “row 3, seat 5” in a cinema, the order matters — $(5; 3)$ and $(3; 5)$ are different points. The axes divide the plane into four quadrants: in quadrant I the signs are $(+; +)$ , in II $(-; +)$ , in III $(-; -)$ , in IV $(+; -)$ . Points on the $O x$ axis have $y = 0$ , those on $O y$ have $x = 0$ .

The concept of a function

A taxi fare depends on the distance: for $x$ km it is $y = 2 x + 5$ thousand soum. If to each value of $x$ there corresponds exactly one $y$ , the relationship is a function; $x$ is the argument (independent variable), $y$ is the value of the function. A function is given by a table, a formula or a graph; the graph is the set of points $(x; f (x))$ . If one $x$ gave two values of $y$ , it would not be a function. The argument takes only meaningful values: a distance cannot be negative ( $x \geq 0$ ).

y = kx — direct proportion

If $x$ doubles, $y$ also doubles: $y = k x$ , $k \neq 0$ . Its graph is a straight line through the origin. $k$ is the gradient: how much $y$ changes when $x$ grows by one unit ( $k = \frac{y}{x}$ is the same at every point, $x \neq 0$ ). If $k > 0$ the line rises to the right, if $k < 0$ it falls; the larger $| k |$ , the steeper the line. Example: in uniform motion the distance is $s = v t$ (physics — the “Metro: from station to station” experiment).

y = kx + b — linear function

If every point of the graph of $y = k x$ is raised by $b$ , we get a parallel straight line — the graph of $y = k x + b$ ; it meets the $O y$ axis at $(0; b)$ . If $k > 0$ the function is increasing, if $k < 0$ decreasing, and if $k = 0$ it is the horizontal line $y = b$ . Two lines with equal $k$ are parallel (no common point); with different $k$ they meet at one point (the solution of a system of equations — the “Systems of equations” experiment). Intersection with $O x$ : $k x + b = 0$ , $x = - \frac{b}{k}$ ( $k \neq 0$ ).

Reading from a graph and modelling

Reservoir: $y = 300 - 20 x$ (litres, minutes). The intersection with $O y$ , $b = 300$ , is the starting volume; the intersection with $O x$ is the moment the reservoir empties: $300 - 20 x = 0$ , $x = 15$ min. From the graph we can read the volume at any moment, and the reverse. Project work: measure a real process (the length of a candle, a phone's charge, a temperature) twice, mark the points, and find $k = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ and $b = y_{1} - k x_{1}$ — a linear model is ready. It works when the rate of change is constant; processes such as the height of a thrown ball are described by a parabola (a quadratic model, Grade 10).

About the model

A real taxi fare is rounded and waiting time is added too; the flow from the reservoir is taken as constant (in reality it slows as the water level falls). The graphs are drawn only over the meaningful interval ( $x \geq 0$ ). On the scene the points “snap” to grid nodes with whole coordinates, so $k$ and $b$ turn out to be rational numbers; in general they can be any real numbers.

Did you know?

René Descartes described the method of coordinates in 1637 in his “Geometry” (an appendix to the “Discourse on Method”) — that is why the system is called “Cartesian coordinates”.
Gottfried Leibniz introduced the word “function” into mathematics in 1673, and Leonhard Euler proposed the notation $f (x)$ in 1734.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.