Turn the line, keep the point

The idea

In y = ax + b, a is the slope and b the y-intercept; changing a while keeping b fixed changes the slope but the y-intercept stays the same.

A stick on a nail

Fix a nail at (0, 3) on the y-axis. Now turn a stick around the nail. Every position of the stick is a line y = ax + 3 with a different a.

When you turn the stick (change a), what do you think stays the same?

• The y-intercept
• The slope
• Nothing

Keep three lines

Change a and press Keep this line. Keep three lines with different a.

Same b, different a

In y = ax + b, a is the slope and b is the y-intercept. If you change a and keep b fixed, the slope changes but the y-intercept stays the same.

y = x + 3, y = 2x + 3 and y = 3x + 3 all have b = 3. They all pass through (0, 3), and each is steeper than the last.

Notes

Change a, keep b: the slope changes, the y-intercept stays the same.

Check yourself

The lines y = 5x + 2 and y = −x + 2 have what in common?

The line y = ax + 4 has a changed from 1 to 3. What is its y-intercept now?

Answer: 4

b is still 4. Changing a only changes the slope.

Which line has the same y-intercept as y = 2x − 3?

Every line y = ax − 1 cuts the y-axis at (0, k). What is k?

Answer: -1

b = −1, whatever a is. All these lines pass through (0, −1).

• The same y-intercept, 2 — correct. Yes. Both have b = 2, so both cut the y-axis at (0, 2).
• The same slope. Their slopes are 5 and −1.
• Both pass through the origin. Both cut the y-axis at (0, 2), not at the origin.
• y = 2x + 3. This one has b = 3. It cuts the y-axis at (0, 3).
• y = 5x − 3 — correct. Yes. Both have b = −3.
• y = 3x. This one has b = 0. It passes through the origin.