A parabola is a U-shaped curve defined by a quadratic function in the form f of x equals a x squared plus b x plus c. The parabola has several key features: the vertex is the turning point, the axis of symmetry is a vertical line through the vertex, and the coefficients a, b, and c control the shape and position. Parabolas appear frequently in real-world applications like projectile motion and satellite dishes.
The leading coefficient 'a' in the function f of x equals a x squared has a dramatic effect on the parabola's appearance. When 'a' is positive, the parabola opens upward like a smile. When 'a' is negative, it opens downward like a frown. The absolute value of 'a' controls the width: larger values create narrower parabolas, while smaller values create wider ones. Watch as we change 'a' from negative three to positive three to see these transformations.
The constant term 'c' in the function f of x equals x squared plus c creates vertical shifts of the parabola. When c is positive, the entire parabola moves upward. When c is negative, it moves downward. The shape of the parabola remains exactly the same - only its vertical position changes. Notice that c represents the y-intercept, the point where the parabola crosses the y-axis. Watch as we vary c from negative four to positive four.
Horizontal shifts are controlled by the parameter h in the vertex form f of x equals x minus h squared. When h is positive, the parabola shifts to the right. When h is negative, it shifts to the left. This might seem counterintuitive at first - subtracting h moves the graph right, while adding h moves it left. The vertex moves horizontally from zero comma zero to h comma zero, but the shape and orientation remain unchanged. The axis of symmetry also moves with the vertex.
Now we combine all transformations using the vertex form f of x equals a times x minus h squared plus k. The parameter a controls stretching and reflection, h creates horizontal shifts, and k creates vertical shifts. Let's start with a basic parabola and apply multiple transformations. First, we'll stretch it vertically, then shift it horizontally and vertically. Finally, we'll demonstrate the example function f of x equals negative two times x plus one squared minus three, which combines reflection, stretching, and both types of shifts.