Linear Functions

Q: In y = 3x − 2, what is y when x = 4?

10 — y = 3×4 − 2 = 12 − 2 = 10.

Q: What is the slope of y = 2x + 5?

2 — In y = ax + b, the slope is a. For y = 2x + 5 the slope is 2.

Q: Which equation has a slope of −3 and a y-intercept of 4?

y = −3x + 4 — Slope a = −3 and y-intercept b = 4, so y = −3x + 4.

Q: What is the slope of the line through (2, 7) and (4, 13)?

3 — a = (13−7)/(4−2) = 6/2 = 3.

Q: Where does the graph of y = −2x + 6 cross the x-axis (the x-intercept)?

x = 3 — Set y = 0: 0 = −2x + 6, so x = 3.

Q: An electricity bill is given by y = 25x + 800, where x is the kWh used. What is the bill for 300 kWh?

¥8,300 — y = 25×300 + 800 = 7500 + 800 = ¥8,300.

Q: At what x-coordinate do y = 4x + 2 and y = −x + 7 cross?

x = 1 — Set them equal: 4x + 2 = −x + 7, so 5x = 5 and x = 1.

Q: Which equation has a slope of 2 and passes through the point (1, 5)?

y = 2x + 3 — Substitute (1, 5) into y = 2x + b: 5 = 2 + b, so b = 3.

Where this is used

A slope and a y-intercept — with just those two numbers you can describe almost any steady change in the world. How much fuel a car burns over a distance, an electricity bill (a base charge plus what you use), a prediction of how your weight will change: all of these are linear functions. Even linear regression, one of the first tools you meet in machine learning, is built on this same idea.

Start gently

The graph of a linear function is a straight line. The slope answers "when x goes up by 1, how much does y go up?" — the same steepness you read off a speed graph in primary school.

Key points

What a Linear Function Is

A function you can write in the form y = ax + b. The number a is called the slope, and b is called the y-intercept. Every time x goes up by 1, y changes by a.

What the Slope Tells You

The slope a is how much y changes when x increases by 1. On a graph it tells you how steep the line is. If a > 0 the line rises as you move right; if a < 0 it falls as you move right.

How to Draw the Graph

1) Plot the y-intercept at (0, b). 2) Use the slope a as your step to plot a second point. 3) Draw a straight line through the two points.

Finding the Equation from Two Points

For a line through the points (x₁, y₁) and (x₂, y₂), the slope is a = (y₂−y₁)/(x₂−x₁). Then substitute either point into y = ax + b to find b.

Feel It in an Electricity Bill

Say a plan charges a base fee of ¥1,000 plus ¥30 for every kWh. Using x kWh costs y = 30x + 1000. The slope is 30 (the price per unit) and the y-intercept is 1000 (the base fee you pay even at zero usage).

See it drawn

Change the slope and the line changes its steepness

All three have the same intercept, 1. They meet on the y-axis; only the slope differs.

Check yourself 10questions

Tap an option to check

Q1 In y = 3x − 2, what is y when x = 4?

Q2 What is the slope of y = 2x + 5?

Q3 Which equation has a slope of −3 and a y-intercept of 4?

Q4 What is the slope of the line through (2, 7) and (4, 13)?

Q5 Where does the graph of y = −2x + 6 cross the x-axis (the x-intercept)?

Q6 An electricity bill is given by y = 25x + 800, where x is the kWh used. What is the bill for 300 kWh?

Q7 At what x-coordinate do y = 4x + 2 and y = −x + 7 cross?

Q8 Which equation has a slope of 2 and passes through the point (1, 5)?

Q9 Which statement about y = −x + 4 is true?

Q10 A car travels at 60 km per hour. Which equation gives the distance y km after x hours?

Jobs that use this

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Analyzes trends in stock prices and economic indicators, and writes reports that guide investment decisions.

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Physicist / Research Scientist$85k

Models physical phenomena such as motion at constant speed with linear functions, then checks those models against experimental data.

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