Which statement correctly describes how a affects the parabola y = a(x − h)^2 + k?

Which statement correctly describes how a affects the parabola y = a(x − h)^2 + k?

• A. It only changes the vertex position.
• B. It changes the axis of symmetry.
• C. Smaller a makes the parabola wider; larger a makes it narrower.
• D. It changes the y-intercept.

Answer: (C)

The value of a controls how stretched the parabola is about the vertex. Since the vertex is at (h, k), changing a does not move the vertex or change the axis of symmetry (which stays at x = h). Instead, it scales how tall the graph is for points away from the vertex: a larger magnitude makes the curve steeper and narrower, while a smaller magnitude makes it flatter and wider. In particular, if 0 < |a| < 1 the parabola is wider, and if |a| > 1 it is narrower. The sign of a determines whether the parabola opens upward or downward, but the width depends on the magnitude of a. So smaller a makes the parabola wider; larger a makes it narrower.

The value of a controls how stretched the parabola is about the vertex. Since the vertex is at (h, k), changing a does not move the vertex or change the axis of symmetry (which stays at x = h). Instead, it scales how tall the graph is for points away from the vertex: a larger magnitude makes the curve steeper and narrower, while a smaller magnitude makes it flatter and wider. In particular, if 0 < |a| < 1 the parabola is wider, and if |a| > 1 it is narrower. The sign of a determines whether the parabola opens upward or downward, but the width depends on the magnitude of a. So smaller a makes the parabola wider; larger a makes it narrower.