Integrated Math 2 / Unit 3: Quadratic Functions
3.3 Transformations of Parabolas
Notes
Start from the parent function $y = x^2$. In $y = a(x - h)^2 + k$:
- $h$ shifts right (if positive) or left (if negative).
- $k$ shifts up (if positive) or down (if negative).
- $a < 0$ reflects over the $x$-axis. $|a| > 1$ stretches vertically (narrower). $0 < |a| < 1$ compresses (wider).
- Example: $y = -(x + 1)^2 - 3$ reflects over the $x$-axis, moves left 1 and down 3.
Practice
How does the graph of $y = (x - 5)^2$ compare to $y = x^2$?
Subtracting inside the parentheses moves the graph right by 5.
Compared to $y = x^2$, the graph of $y = 0.25x^2$ is:
Since $0 < 0.25 < 1$, the parabola is vertically compressed, which makes it wider.
Which describes $y = -(x + 1)^2 - 3$ compared to $y = x^2$?
The negative reflects it, $+1$ inside moves it left 1, and $-3$ outside moves it down 3.
Ready for more? Take the Unit 3 test.