In the vertex form y = a(x − h)^2 + k, what does the parameter a determine?

• A. It determines the vertex coordinates.
• B. It determines the axis of symmetry.
• C. It determines the y-intercept.
• D. It determines the width and direction of the parabola.

Answer: (D)

The a in this form controls how stretched or squished the parabola is and which way it opens. It scales the squared term, so changing a changes the width: larger magnitude |a| makes the graph narrower, while smaller magnitude |a| (but still nonzero) makes it wider. The sign of a decides the direction: positive a makes the parabola open upward, negative a makes it open downward. The vertex itself is determined by h and k, not a, so the position stays the same while the shape changes. The axis of symmetry is the line x = h. Note that the y-intercept can be affected by a as well, since y when x = 0 equals a(h)^2 + k, but the main effect of a is width and direction.