Question

Use a graphing utility to graph the parabolas $y^2=4px,$ 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: As the value of $p$ changes, the direction, vertex, and focal length of the parabolas are affected. The vertex remains at the origin, while the direction and focal length vary based on the value of $p$. If $p>0$, the parabolas open towards the right, and if $p<0$, they open towards the left. The focal length depends on the absolute value of $p$; the larger the absolute value, the wider the parabola, and the smaller the absolute value, the narrower the parabola.

Step-by-step solution

Understand the equation of the parabola

The equation given is $y^2=4px$, which represents a parabola with its vertex at the origin (0,0) and the focus at $(p,0)$. Since $p$ can vary, the position of the focus changes. The focal length of the parabola is the distance between the vertex and the focus, which is equal to $|p|$. If $p>0$, then the parabola is opening towards right; and if $p<0$, then the parabola is opening towards left.

Graphing the parabolas using a graphing utility

Use a graphing utility (like Desmos, Geogebra, or even a graphing calculator) to graph the parabolas for each of the given values of $p$ (-5, -2, -1, 1, 2, 5). After graphing these parabolas, you should have a visualization of how these parabolas are arranged on the coordinate plane.

Analyzing the shapes of the graphs

Observe the graphs created in Step 2. You'll notice the following: - For $p>0$ (1,2,5), the parabolas open to the right, with their vertex at the origin (0,0). - For $p<0$ (-5, -2, -1), the parabolas open to the left, with their vertex at the origin (0,0). - The focal length (the distance between the vertex and the focus) affects the stretching or compressing of the parabolas. Larger absolute value of $p$ results in a more stretched (wider) parabola, whereas smaller absolute value of $p$ results in a more compressed (narrower) parabola.

Conclusion

As $p$ changes, the parabolas' direction, vertex, and focal length are affected. The vertex remains at the origin, while the direction and the focal length vary based on the value of $p$. If $p>0$, the parabolas open towards the right, and if $p<0$, they open towards the left. The focal length depends on the absolute value of $p$; the larger the absolute value, the wider the parabola, and the smaller the absolute value, the narrower the parabola.

Key concepts

Graphing Utility

When exploring the graph of a parabola given by the equation $y^2=4px$, a graphing utility can be an invaluable tool. Graphing utilities like Desmos or Geogebra allow you to visualize complex equations quickly and accurately. Instead of plotting individual points manually, these utilities plot the entire curve, giving a clear picture of its shape and orientation.

To graph this particular parabola, input the equation into your chosen utility. You can then change the value of $p$ to see how it affects the parabola's appearance. The graphing utility will dynamically adjust the graph to show any changes in orientation and stretch. This visual approach helps in understanding not only the opening direction of the parabola but also how the width varies with different values of $p$.

• Right-opening parabolas occur when $p>0$.
• Left-opening parabolas occur when $p<0$.
• A larger absolute value of $p$ indicates a wider parabola.
• A smaller absolute value of $p$ results in a narrower parabola.

Vertex of a Parabola

The vertex of a parabola defined by $y^2=4px$ is always at the origin, $(0,0)$, based on this specific form of the equation. The vertex acts as a pivotal point from which the parabola opens. In this case, it serves as the point where the curve is closest or tightest before it begins to spread outward.

Understanding the location of the vertex is crucial because it helps in symmetry analysis. Parabolas have a vertical or horizontal axis of symmetry, which can be determined by the vertex in the context of this equation. Here, the vertex not only provides location but also serves as the center of symmetry around which the parabola opens either to the left or right.

• Vertex at $(0,0)$ means the curve is poised symmetrically around this point.
• Knowing the vertex simplifies graphing and comparison tasks.
• It remains unchanged even as $p$ adjusts during graphing.

Focal Length of a Parabola

In parabolas described by the equation $y^2=4px$, the focal length is very important. It is the distance from the vertex $(0,0)$ to the focus $(p,0)$. This measurement directly influences the parabola's shape. Here, $|p|$ gives the focal length, regardless of whether $p$ is positive or negative.

The focal length plays a key role in how "wide" or "narrow" the parabola appears. A larger focal length, indicated by a greater absolute value of $p$, results in a wider, more gently curving parabola. Conversely, a smaller focal length, noticed when the absolute value of $p$ is smaller, causes a steeper, narrower curve.

• Focal length is the absolute value of $p$.
• Absolute value provides consistency in understanding fairness regardless of parabolas' orientation.
• Wider parabolas have larger focal lengths, while narrower parabolas have smaller ones.