Question

Sketch the graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work. $$x^{2}=12 y$$

Short answer

Question: Determine the key attributes of the parabola $$x^2 = 12y$$, including its vertex, focus, and directrix. Then, sketch the graph. Answer: The parabola has a vertex at the origin (0,0), a focus at point (0,3), and a directrix at $$y = -3$$. To sketch the graph, plot the vertex, focus, and directrix, and draw the parabolic shape accordingly.

Step-by-step solution

Rewrite the equation in standard form

To rewrite the equation $$x^2 = 12y$$ in the standard form, we can compare it with the standard form of a parabola. Since the x term is squared, the parabola will be vertically oriented. The equation for a vertically oriented parabola with the vertex at the origin is $$x^2 = 4py$$, where p is the distance between the vertex and the focus. In our equation, we have $$x^2 = 12y$$, which means $$4p = 12$$. Solving for p, we get: $$p = \frac{12}{4} = 3$$ Now, we can rewrite the given equation as $$x^2 = 4(3)y$$. The vertex of the parabola is at the origin (0,0).

Find the focus

With the value of p = 3, we can now find the focus of the parabola. Since the parabola is vertically oriented, the focuswill be above or below the vertex. Specifically, it will be at the point (h, k + p), where (h, k) is the vertex of the parabola. Since the vertex is located at (0,0), the focus will be at $$\left(0, 0 + 3\right) = \left(0,3\right)$$. So, the focus of the parabola is at point (0, 3).

Find the directrix

The directrix of a vertically oriented parabola with vertex at (0,0), will be a horizontal line situated below the vertex. The equation of a horizontal line is given by $$y = c$$, where c is a constant. Specifically, the directrix will be at y = -p. In our case, p = 3, so the directrix will be at $$y = -3$$.

Sketch the graph

Now, we have all the required information to sketch the graph of the parabola $$x^2 = 12y$$. We'll plot the vertex (0,0), the focus (0,3), and the directrix at $$y=-3$$, and viola, we'll have our parabola.

Check using a graphing utility

To verify our work, we can use a graphing utility like Desmos or any other similar software. Simply input the equation $$x^2 = 12y$$ and check if the graph matches our analysis. After verifying, we'll find that the graph does indeed have a vertex at the origin (0,0), a focus at (0,3), and a directrix at $$y = -3$$.

Key concepts

Focus of a Parabola

The focus of a parabola is a special point located inside the curve. It's one of the defining characteristics of the parabola. You can think of it as the "target" that all points on the parabola are trying to reflect towards.
Every parabola has a single focus that works in harmony with something called the directrix. Together, the focus and directrix define the shape and orientation of the parabola.
In a vertically oriented parabola with an equation similar to \(x^2 = 4py\), the focus sits at \((h, k + p)\). Here, \((h, k)\) is the vertex, which is the "tip" of the parabola. The "p" represents the distance from the vertex to the focus.

• For the equation \(x^2 = 12y\), the focus is calculated by first finding \(p\) from \(x^2 = 4py\), where \(4p = 12\), giving \(p = 3\).


• The vertex is at \((0,0)\), so adding \(p\) gives us the focus at \((0,3)\).

This point (0, 3) is where all the activity of the parabola converges.

Directrix of a Parabola

The directrix of a parabola is a critical straight line that is perpendicular to the axis of the parabola. It's positioned on the opposite side of the vertex from the focus.
While the focus pulls the parabola towards its location, the directrix helps to guide its curve. In simple terms, every point on the parabola is equidistant from both the focus and this line.
For a vertical parabola, the directrix is a horizontal line and its equation is of the form \(y = c\). Specifically, it is located at \(y = k - p\) for a parabola opening upwards.

• In the case of the equation \(x^2 = 12y\), since \(p = 3\), the directrix lies at \(y = -3\), which is located below the vertex \((0,0)\).

This line is essential for maintaining the parabola's balanced structure.

Standard Form of a Parabola

Understanding the standard form of a parabola is crucial for sketching and identifying key features like the vertex, focus, and directrix.
The standard form helps us quickly identify the "recipe" of the parabola. For vertically oriented parabolas, the standard form is \(x^2 = 4py\), where \(p\) defines the distance from the vertex to the focus.
In this case, the "4p" part tells us how "open" or "narrow" the parabola is.

• In the exercise, the equation \(x^2 = 12y\) is compared to \(x^2 = 4py\), helping us find \(p = 3\).


• This indicates a relatively "normal" opening, as \(p\) isn't too large or small.

Understanding this form not only helps in sketching the graph but also in predicting how slight changes in \(p\) could alter the parabola's shape.