Algebra
Linear functions
Transformations of linear functions
20 practice questions
2 video lessons
Theory + worked examples
Theory
The parent line \(y=x\) transforms in a few ways:
- \(y=x+k\): shifts up (or down) by \(k\).
- \(y=mx\): changes the steepness (slope).
- \(y=-x\): reflects over the \(x\)-axis.
Changing \(b\) slides the line; changing \(m\) tilts it.
\(y=x\), a shift, and a reflection.
Transformations of \(y=x\).
The general form:
\[y=mx+b\]
\(m\) controls tilt and reflection; \(b\) controls the shift.
How to describe a transformation
- Compare the slope \(m\) with the parent's.
- A negative \(m\) is a reflection.
- Compare the intercept \(b\) for the shift.
- Combine the effects.
Example 1 — Vertical shift
Describe \(y=x+3\) compared with \(y=x\).
Solution
\(+3\) shifts the line up.
| \(y=x+3\) | \(\Rightarrow\) | \(\text{up } 3\) |
Example 2 — Steeper line
How does \(y=2x\) compare with \(y=x\)?
Solution
A larger slope is steeper.
| \(y=2x\) | \(\Rightarrow\) | \(\text{twice as steep}\) |
Example 3 — Reflection
Describe \(y=-x\) compared with \(y=x\).
Solution
The negative slope reflects it over the \(x\)-axis.
| \(y=-x\) | \(\Rightarrow\) | \(\text{reflection}\) |
Example 4 — Combined
Describe \(y=2x-1\) from \(y=x\).
Solution
Steeper (slope \(2\)) and shifted down \(1\).
| \(y=2x-1\) | \(\Rightarrow\) | \(\text{steeper, down } 1\) |
Common pitfalls
\(b\) shifts vertically; \(m\) changes steepness.
A negative slope reflects the line.
Larger \(|m|\) is steeper, not taller.
Frequently asked questions
What does \(y=x+3\) do to \(y=x\)?
Shifts it up \(3\).
What does a larger slope do?
Makes the line steeper.
What does \(y=-x\) represent?
A reflection of \(y=x\) over the \(x\)-axis.
Which part shifts the line?
The constant \(b\).
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