Written by Sunny Yoon

Examine graphs for the parabola for different values of a, b, and c. y = ax^2 + bx + c

(a, b, c can be any rational numbers). Try using the GC animation by replacing a, b, or c with an n and selecting an appropriate range for n.

I'm going to change the values for a. First of all, a cannot be zero because if a is zero then it becomes a linear function.

As n goes from -5 to 5, the y-intercept at (0,1) doesn't change, but the shape of the parabola and the x-intercept. When n < 0, the parabola becomes an upside down "U" shape. The vertex of the parabola is the maximum point. When n >0, the parabola becomes a "U" shape. The vertex is the minimum point. Although the y-intercept stays the same at (0,1), x-intercept changes according to a different value of n. When n < 0 (closer to -5), there appears to be 2 x-intercepts. However, when n >0 (closer to 5), there appears to be no x-intercepts.

What happens when you vary the 'b'? Using y = -2x^2 + nx + 1, where n varies from -5 to 5, the following animation will demonstrate how the parabola changes as b varies.

Since a < 0, the vertex is a maximum point of this parabola. Also, there are always 2 x-intercepts in this parabola. The y-intercept stays the same at (0, 1). However, when n = 0, the y-intercept becomes the vertex. Although there are always 2 x-intercepts, the value of the x-intercept changes as n varies.

What happens when you vary the 'c'? Graph y = x^2 - 4x + c

The general shape of the parabola doesn't change. Since a > 0, the vertex is a minimum point of the parabola. However, unlike the previous two examples, the y-intercept changes as n varies. Also, x-intercept varies as n varies.

In conclusion, when you vary 'a' from a quadratic function, the general shape of the parabola changes. The vertex will either be the minimum or maximum depending on 'a'. y-intercept doesn't change at all. When you vary 'b' from a quadratic function, the general shape of the parabola and the y-intercept don't change at all. However, x-intercepts vary depending on 'b'. When you vary 'c' from a quadratic function, y-intercept and x-intercepts change depending on 'c'. The general shape of the parabola won't change.

2. Fix two of the values for a, b, and c. Make at least 5 graphs on the same axes as you vary the third value.