How does the coefficient of x^2 affect the shape and position of the parabola in a quadratic function?

Answers:

• In a quadratic function of the form ( f(x) = ax^2 + bx + c ), the coefficient ( a ) (which is the coefficient of ( x^2 )) plays a crucial role in determining the shape and position of the parabola. Here's how it affects the graph:

  1. Direction of Opening:

     • If ( a > 0 ): The parabola opens upward, resembling a "U" shape.
     • If ( a < 0 ): The parabola opens downward, resembling an "n" shape.

  2. Width of the Parabola:

     • The absolute value of ( a ) (denoted as ( |a| )) affects how "wide" or "narrow" the parabola is.
     • If ( |a| > 1 ): The parabola is narrower, which means it rises or falls more steeply.
     • If ( |a| < 1 ): The parabola is wider, indicating a more gradual rise or fall.

  3. Vertex Position:

     • While the coefficient ( a ) itself does not directly determine the vertex location, it does influence the position of the vertex in conjunction with the other coefficients ( b ) and ( c ). The vertex can be found using the formula ( x = -\frac{b}{2a} ). The y-coordinate of the vertex can be found by substituting this x-value back into the function.

  4. Symmetry:

     • The parabola is symmetric about its vertex. The coefficient ( a ) does not affect the axis of symmetry directly, but since it helps define the vertex position, it can indirectly influence the overall shape.

  In summary, the coefficient ( a ) not only determines the direction in which the parabola opens but also affects its width. Understanding the value and sign of ( a ) helps predict the overall behavior of the quadratic function's graph.