Algebra
Quadratic equations and functions
Transformations of quadratic functions
20 practice questions
2 video lessons
Theory + worked examples
Theory
The parent parabola \(y=x^2\) transforms through vertex form \(y=a(x-h)^2+k\):
- \(h\): horizontal shift (right if \(h>0\)).
- \(k\): vertical shift (up if \(k>0\)).
- \(a<0\): reflects (opens down).
- \(|a|>1\): narrower; \(|a|<1\): wider.
Inside the square affects \(x\) (opposite way); outside affects \(y\).
\((x-1)^2+2\) is \(y=x^2\) shifted right 1, up 2.
Transformations of \(y=x^2\).
The template:
\[y=a(x-h)^2+k\]
\((x-h)\) shifts right by \(h\).
How to describe a transformation
- Read \(h\) and \(k\) for the shifts.
- Check the sign of \(a\) for a reflection.
- Check \(|a|\) for stretch or compression.
- Combine the effects.
Example 1 — Horizontal shift
Describe \(y=(x-3)^2\) compared with \(y=x^2\).
Solution
\(h=3\) shifts right.
| \((x-3)^2\) | \(\Rightarrow\) | \(\text{right } 3\) |
Example 2 — Vertical shift
Describe \(y=x^2+4\).
Solution
\(+4\) shifts up.
| \(x^2+4\) | \(\Rightarrow\) | \(\text{up } 4\) |
Example 3 — Reflection
Describe \(y=-x^2\).
Solution
The negative reflects it over the \(x\)-axis (opens down).
| \(-x^2\) | \(\Rightarrow\) | \(\text{opens down}\) |
Example 4 — Combined
Describe \(y=(x-1)^2+2\).
Solution
Right \(1\), up \(2\) — vertex \((1,2)\).
| \(\text{vertex}\) | \(=\) | \((1,2)\) |
Common pitfalls
\((x-h)\) shifts right, not left.
A negative \(a\) opens the parabola down.
\(|a|>1\) is narrower, not taller.
Frequently asked questions
What does \(y=(x-2)^2\) do to \(y=x^2\)?
Shifts it right \(2\).
What does a negative \(a\) do?
Reflects the parabola to open downward.
What does \(+k\) do?
Shifts the parabola up by \(k\).
What makes a parabola narrower?
A larger \(|a|\).
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