- Explain what a function is
- Plot points on the coordinate plane
- Draw the graph of y = kx + b
- Explain what the coefficients k and b mean
A mobile operator charges a 5-manat monthly fee plus 2 manat for every gigabyte of data. If you use x GB a month, you pay y = 2x + 5 manat. Each value of x gives exactly one value of y — this is a function. And its graph is a straight line.
Functions and the coordinate plane
A rule that assigns exactly one value of y to each value of the variable x (the input, or argument). y is called the value of the function and is written y = f(x).
To picture a function we use the coordinate plane: the horizontal x-axis and the vertical y-axis. Each point has two coordinates: to reach A(2, 3), go 2 units right from the origin and 3 units up. The graph of a function is the set of all points (x, f(x)).
| x (GB) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y = 2x + 5 (manat) | 5 | 7 | 9 | 11 | 13 |
The linear function y = kx + b
- kthe slope: how much y changes when x increases by 1
- bwhere the graph crosses the y-axis: (0, b)
The graph of a linear function is a straight line, so two points are enough to draw it. Some textbooks write the formula as y = mx + b, and in Türkiye as y = mx + n.
- If k > 0, the function is increasing: the graph rises from left to right.
- If k < 0, the function is decreasing: the graph falls.
- If k = 0, y = b is a horizontal line parallel to the x-axis.
- If b = 0, y = kx is direct proportion, and the graph passes through the origin.
- Lines with the same k are parallel.
Draw the graph of y = −2x + 4. Does the point A(3, −2) lie on it?
Show solutionHide solution
x = 0: y = 4 ⇒ (0, 4) — where it crosses the y-axis.
y = 0: −2x + 4 = 0 ⇒ x = 2 ⇒ (2, 0) — where it crosses the x-axis.
Draw the line through these points; since k = −2 < 0, the function is decreasing.
A(3, −2): −2 · 3 + 4 = −2 — true, so A lies on the graph.
The graph of a linear function passes through (1, 3) and (3, 7). Write the formula of the function.
Show solutionHide solution
k = 4 ÷ 2 = 2.
From the point (1, 3): 3 = 2 · 1 + b ⇒ b = 1.
Answer: y = 2x + 1. Check: 2 · 3 + 1 = 7.
Plan A: y = 2x + 5, plan B: y = 3x + 1 (x in GB, y in manat). For what amount of data do the plans cost the same, and which is better for someone who uses little data?
Show solutionHide solution
At x = 4 GB both plans cost 13 manat — this is where the graphs cross: (4, 13).
For example, at x = 1: A costs 7 manat, B costs 4 manat. So plan B is cheaper for less than 4 GB, and A for more.
Key points
- A function assigns exactly one y to each x.
- The graph of y = kx + b is a straight line; two points are enough to draw it.
- k > 0 — increasing, k < 0 — decreasing; b is where the line meets the y-axis.
- Lines with equal slopes are parallel.
Check yourself
10 questions. Every correct answer earns XP.