Question

lets investigate different equations to understand the relationships between the structure of the equation and its graph.
consider the parabola represented by the equation ( x ^ { 2 } = - 12 y ).
activity 4: using the constant p to graph a parabola
identify the coordinates of the vertex
what is the value of p?
write the equation of the axis of symmetry.
determine the coordinates of the focus
write the equation of the directrix.
the parabola is concave _.

Explanation:

Step1: Recall the standard form of parabola equation

The standard form of a parabola equation is \(x^{2}=4py\).

Step2: Compare with the given equation

Given \(x^{2}=-12y\), comparing with \(x^{2}=4py\), we have \(4p=-12\).

Step3: Solve for \(p\)

Divide both sides of \(4p = - 12\) by \(4\), we get \(p=\frac{-12}{4}=-3\).

Step4: Find the vertex

For the parabola \(x^{2}=4py\), the vertex is \((0,0)\).

Step5: Find the focus

The focus of the parabola \(x^{2}=4py\) is \((0,p)\). Since \(p=-3\), the focus is \((0, - 3)\).

Step6: Find the axis of symmetry

For the parabola \(x^{2}=4py\), the axis of symmetry is \(x = 0\).

Step7: Determine the concavity

Since \(p=-3<0\), the parabola is concave down.

Step8: Find the directrix

The equation of the directrix of the parabola \(x^{2}=4py\) is \(y=-p\). Since \(p = - 3\), the directrix is \(y = 3\).

Answer:

• Value of \(p\): \(-3\)
• Vertex: \((0,0)\)
• Focus: \((0,-3)\)
• Axis of symmetry: \(x = 0\)
• Concavity: down
• Directrix: \(y = 3\)