Quadratic functions

In y = a(x − h)² + k with a ≠ 0, the parabola’s vertex is (h, k): h sets its horizontal position and k its vertical position. Positive a opens upward and negative a downward; larger |a| makes the parabola narrower. When a = 0, the function is the horizontal line y = k with no unique vertex. Adjust one parameter at a time to see each effect.

Questions

How does the vertex move when you change only the horizontal shift h?

How do the sign and magnitude of a affect the parabola's direction and width?

How can you choose a, h, and k to place the vertex at (2, −1) with the parabola opening downward?

Predict

Before touching the sliders, predict whether changing h from 0 to 2 moves the vertex left or right. Compare your prediction with the result.

Try

Keep a = 1 and k = 0, and change only h. Then return h to zero and adjust k and a separately. The blue dashed curve always shows y = x²; the amber solid curve shows the current function.

Observe

h sets the vertex's horizontal position; k sets its vertical position. A larger absolute value of a makes the parabola narrower. Negative a makes it open downward. At a = 0, the graph becomes a horizontal line.

Explain

Why does positive h move the parabola to the right?

For a ≠ 0, the squared term in y = a(x − h)² + k vanishes at x = h, giving the vertex (h, k). When h changes from 0 to 2, the vertex moves to x = 2, to the right. More generally, reproducing an old value of x² requires increasing the input by h. Horizontal shifts change the input; vertical shifts change the output.

How does a change the opening and width?

For the same nonzero distance from x = h, multiplying by a scales the vertical distance from y = k. Larger |a| therefore makes the parabola narrower at a fixed vertical distance from the vertex. Negative a reflects it downward. At a = 0, every input gives y = k: this is a constant function with no unique vertex, not a parabola.

Check

Move the vertex to (2, −1) and make the parabola open downward: h = 2, k = −1, a < 0. Do pairs of points symmetric about x = h have equal function values?

Explore further

Do the same transformations work for the absolute-value function? For any function f, y = a f(x − h) + k follows similar rules for translation and vertical scaling.

Continue with Derivatives and tangents: how does the slope between two points approach the slope at one point? In that experiment, h means the gap between inputs, not a horizontal shift.