Linear function

Explanation

In y = ax + b, a is the slope (in Dutch: richtingscoëfficiënt): how much y changes when x goes up by 1. b is where the line crosses the y-axis. At havo and vwo you can switch on a second line to see where two lines meet, or that parallel lines never do.

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What does a really mean?

a is how much y changes when x goes up by one. In y = 3x − 5 the line climbs 3 for every step to the right. With a negative a, each step to the right takes you down, so the line falls. A negative slope does not mean the line “goes left”. b is the value of y when x = 0, which is where the line meets the y-axis.

Examples by level

• vmbo: renting a bike costs €4 plus €3 per hour, so cost = 3 · hours + 4. For 5 hours that is 3 · 5 + 4 = 19 euros. The 3 is the slope and the 4 is the starting amount.
• havo: a line through (2; 1) and (5; 10) has slope a = (10 − 1) ÷ (5 − 2) = 9 ÷ 3 = 3. Using (2; 1): b = 1 − 3 · 2 = −5, which gives y = 3x − 5.
• vwo: where do y = 3x − 5 and y = −2x + 5 meet? Set them equal: 3x − 5 = −2x + 5, so 5x = 10 and x = 2. Then y = 3 · 2 − 5 = 1, so they meet at (2; 1).

Parallel or crossing?

Compare the slopes before anything else. Different slopes mean the lines cross at exactly one point. Equal slopes with different b values mean parallel lines that never meet. If a and b are both equal, it is really one line drawn twice. Turn on the second line in the simulation and make each case yourself.

Tasks

1. Make the line pass through the point (2; 5).
2. Turn on “Show second line” and make two parallel lines.
3. Make the two lines intersect at x = 1.
4. Make a line through (0; 4) that goes 2 down for every step to the right.

Common mistakes

Test yourself

Practice with linear equations.

Work it out

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