Transformations of Quadratic Functions

Parabola Translations

Ever wondered how to move a parabola around on a graph? It's actually pretty simple! The parent quadratic function f$x$ = x² can be shifted horizontally or vertically using the formula g$x$ = a$x-h$² + k.

When you see $x-h$ in a function, it creates a horizontal shift. If h is positive, the parabola moves right; if h is negative, it moves left. For example, $x-3$² shifts the parabola 3 units right, while $x+4$² shifts it 4 units left (since x+4 = x-$-4$).

For vertical shifts, just look at the value of k added at the end. A positive k pushes the parabola upward, while a negative k pulls it downward. So x² + 5 moves the parabola up 5 units, and x² - 2 moves it down 2 units.

Quick Tip: Remember "h" for horizontal and "k" for... well, not horizontal! A good way to remember the direction: when h is positive, the parabola moves right (in the positive direction of the x-axis).

Reflections, Stretches, and Shrinks

Parabolas can do more than just move around—they can flip and change shape too! When you put a negative sign in front of the entire function (-f$x$ = -x²), the parabola reflects over the x-axis, flipping upside down.

Interestingly, replacing x with -x in f$x$ = x² doesn't change the parabola's appearance. This is because $-x$² = x², making the parabola symmetrical about the y-axis.

You can stretch or shrink a parabola horizontally by replacing x with ax. When 0 < a < 1 (like ½), the parabola stretches wider. When a > 1 (like 2), it shrinks narrower. For example, f(2x) = (2x)² creates a narrower parabola than the original.

For vertical stretches and shrinks, multiply the entire function by a constant a. When a > 1, the parabola stretches taller; when 0 < a < 1, it shrinks shorter. So 3x² is stretched vertically compared to x², while ½x² is shrunk.

Remember: Horizontal transformations work oppositely from what you might expect—multiplying by a smaller fraction (like ½) makes the parabola wider, not narrower!