Question

question 4 of 25

which of the following equations will produce the graph shown below?

a. \\(x^2 = -8y\\)

b. \\(x - \frac{1}{8}y^2 = 0\\)

c. \\(x^2 = 8y\\)

d. \\(y = \frac{1}{2}x^2\\)

Explanation:

Identify key features from the graph

The graph is a vertical parabola opening upwards with its vertex at the origin \((0,0)\).
A blue point representing the focus is plotted at \((0, 2)\).

Determine the standard equation of the parabola

Using the Parabola Definition, a vertical parabola opening upwards with vertex at \((0,0)\) and focus at \((0, p)\) has the standard equation:

$$x^2 = 4py$$

Calculate the equation parameters

Since the focus is at \((0, 2)\), we have \(p = 2\).
Substituting \(p = 2\) into the standard equation:

$$x^2 = 4(2)y \implies x^2 = 8y$$

Answer:

• A. \(x^2 = -8y\)
• B. \(x - \frac{1}{8}y^2 = 0\)
• C. \(x^2 = 8y\) (Correct answer)
• D. \(y = \frac{1}{2}x^2\)