The linear function and its graph

Imagine a taxi ride. You pay 500 tenge to get in, and then 100 tenge for every kilometre. How much is a 3 km ride? A 7 km ride?

Call the distance x and the fare y. Then the fare follows one rule for any distance:

y = 100x + 500

Put in x = 3: y = 100 · 3 + 500 = 800. Put in x = 7: y = 1200. Every distance gets exactly one fare. A rule like this is called a function. Here x is the independent variable, or argument, and y is the dependent variable, or value of the function.

Take away the specific numbers and keep only the shape of the formula. A linear function is a function that can be given by the formula

y = kx + b,

where k and b are numbers and x is the independent variable. The number k is called the slope, and the number b the constant term. In the taxi formula k = 100 and b = 500.

The numbers k and b can be anything: negative, fractional, even zero. Here are some linear functions and their coefficients:

• y = 3x − 2: k = 3, b = −2
• y = −x + 5: k = −1, b = 5 (there is a minus in front of x, so k = −1)
• y = 4 − 3x: k = −3, b = 4 (the order of the terms does not matter, k is always the number next to x)
• y = 0.5x: k = 0.5, b = 0
• y = 7: k = 0, b = 7

The graph of a function is the set of all points on the coordinate plane whose x-coordinate and y-coordinate are linked by the formula of the function. The graph of a linear function is always a straight line.

Exactly one straight line passes through two points. So to draw the graph it is enough to find two points and then draw a line through them with a ruler.

Let us draw the graph of y = 2x − 1.

1. Choose two convenient values of x. Zero is always convenient: at x = 0 we get y = 2 · 0 − 1 = −1. Take x = 2 as well: y = 2 · 2 − 1 = 3.
2. Mark the points (0, −1) and (2, 3) on the coordinate plane.
3. Draw a straight line through them.

To check yourself, take a third point: at x = 1 we get y = 1. The point (1, 1) must lie on the same line.

How can you tell whether a point lies on the graph without drawing it? Substitute the coordinates of the point into the formula. If the equality is true, the point belongs to the graph.

• The point (3, 5) and the function y = 2x − 1: 2 · 3 − 1 = 5. The equality is true, the point is on the graph.
• The point (1, 2) and the same function: 2 · 1 − 1 = 1, not 2. The point is not on the graph.

The same idea finds a missing coordinate. If the point (5, y) lies on the graph of y = 2x − 1, then y = 2 · 5 − 1 = 9.

The number b shows where the line crosses the y-axis. Put x = 0 into y = kx + b and you get y = b. So the point (0, b) always lies on the graph. If you change b, the line slides up or down without turning.

The number k shows how fast y grows. When x increases by 1, y changes by exactly k.

• If k > 0, the line goes up from left to right: the function increases.
• If k < 0, the line goes down: the function decreases.
• If k = 0, the formula becomes y = b: a horizontal line, parallel to the x-axis.

The larger k is in absolute value, the steeper the line. If you change k, the line turns around the point (0, b).

The crossing point with the x-axis can also be found without a drawing. On the x-axis the y-coordinate is zero, so we solve kx + b = 0. For example, for y = 2x − 6: 2x − 6 = 0, x = 3. The line crosses the x-axis at (3, 0) and the y-axis at (0, −6).

A special case: if b = 0, the formula becomes y = kx. This relationship is called direct proportionality, and its graph passes through the origin (0, 0).

Take two linear functions: y = k₁x + b₁ and y = k₂x + b₂. How are their graphs placed? The coefficients decide everything.

• k₁ = k₂ and b₁ ≠ b₂: the lines are parallel. They rise equally steeply but cross the y-axis at different points, so they have no common points. For example, y = 2x + 1 and y = 2x − 3.
• k₁ ≠ k₂: the lines intersect in exactly one point, whatever b₁ and b₂ are.
• k₁ = k₂ and b₁ = b₂: it is the same line, the graphs coincide.

Note that equal values of b do not make lines parallel. They only mean that both lines pass through the same point on the y-axis.

How do you find the intersection point without a drawing? At that point both lines have the same x and the same y. So the values of the functions are equal: set the formulas equal and solve the equation.

Example: y = 2x + 1 and y = −x + 4.

1. 2x + 1 = −x + 4
2. 3x = 3, so x = 1
3. Put it into either formula: y = 2 · 1 + 1 = 3

The intersection point is (1, 3). Check with the second formula: −1 + 4 = 3. It matches.

• A linear function is given by y = kx + b, where k and b are numbers. Its graph is a straight line.
• Two points are enough to draw the graph. A point belongs to the graph if its coordinates turn the formula into a true equality.
• b shows where the line crosses the y-axis: at (0, b). k shows the direction and steepness: for k > 0 the function increases, for k < 0 it decreases, for k = 0 the line is horizontal.
• Equal k: the lines are parallel (or the same line if b is equal too). Different k: the lines intersect in one point, found by setting the formulas equal.