Question

Use a graphing utility to graph the parabolas \(y^{2}=4 p x,\) for \(p=-5,-2,-1,1,2,\) and 5 on the same set of axes. Explain how the shapes of the curves vary as \(p\) changes.

Short answer

Answer: The value of p affects the shape and orientation of the parabolas in several ways: (1) Positive values of p result in parabolas opening towards the right, while negative values of p result in parabolas opening towards the left. (2) The vertex of all parabolas is at the origin (0,0). (3) Parabolas with larger magnitudes of p will appear taller and narrower, while those with smaller magnitudes of p will appear shorter and wider. (4) The value of p is related to the focal distance, controlling the sharpness or "flatness" of the curve.

Step-by-step solution

Substitute the values of p and create the parabola equations.

Let's substitute the values of \(p\) into the given equation \(y^2 = 4px\) to create 6 parabola equations: \(p = -5: y^2 = -20x\) \(p = -2: y^2 = -8x\) \(p = -1: y^2 = -4x\) \(p = 1: y^2 = 4x\) \(p = 2: y^2 = 8x\) \(p = 5: y^2 = 20x\)

Graph the parabolas

Use a graphing utility, such as a graphing calculator or an online graphing tool, to plot each parabola equation on the same set of axes. Be sure to label each parabola with the corresponding values of \(p\) for clarity.

Analyze the shape variations

Observe how the shape of the parabolas change as \(p\) changes. Here are some notable characteristics: 1. For positive values of \(p\), the parabolas open towards the right. Conversely, for negative values of \(p\), the parabolas open towards the left. 2. For all values of \(p\), the parabolas' vertex is at the origin (0,0). 3. Parabolas with larger magnitudes of \(p\) will stretch more in the vertical direction, making them appear taller and narrower. Parabolas with smaller magnitudes of \(p\) will stretch less in the vertical direction, making them appear shorter and wider. 4. The value of \(p\) is related to the focal distance of the parabola. The focal distance \(|p|\) is the distance from the vertex to the focus, and it controls the sharpness or "flatness" of the curve. By analyzing the relationship between the values of \(p\) and the shapes of the curves, it is evident that the value of \(p\) has a significant impact on the appearance and orientation of the parabolas.