Problem 57 - Algebra 2

Question

Multiple Choice The focus of a parabola is at \((-3,3) .\) The vertex is at \((0,3) .\) What is the equation of the parabola? A \(y-3=\frac{1}{12} x^{2}\) B \(x=-\frac{1}{12}(y-3)^{2}\) C \(y=-\frac{1}{12}(x-3)^{2}\) D \(x=\frac{1}{12}(y+3)^{2}\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x=-\frac{1}{12}(y-3)^{2}\). So, option B is correct.

Step 1: Determine the Direction of the Parabola

The direction the parabola opens is determined by the positions of the focus and the vertex. The parabola always opens in the direction of the focus. If the focus is to the left of the vertex, the parabola opens leftwards. Here, the focus is at (-3, 3) and the vertex is at (0, 3), so the parabola opens to the left.

Step 2: Select the Correct Generic Parabola Equation

Because the parabola opens to the left, the equation should be of the form \(x=A(y-k)^{2}+h\) where (h, k) is the vertex of the parabola and A is a constant that will be determined. The vertex of this parabola is at (0,3), so the equation becomes \(x=A(y-3)^{2}\).

Step 3: Calculate the Constant A

The distance from the vertex to the focus is equal to 1/4A. The distance between the vertex at (0,3) and the focus at (-3,3) is 3 units. Therefore, 1/4A=3, solving for A gives \(A=-\frac{1}{12}\).

Step 4: Substitute A into the Equation

Substituting the calculated A into our equation gives \(x=-\frac{1}{12}(y-3)^{2}\).

Key Concepts

Focus and vertex

Understanding the focus and vertex of a parabola is crucial in determining its graph and equation. The **vertex** of the parabola is the turning point and often represents either the maximum or minimum point, depending on the parabola's orientation. In the case given, the vertex is located at the point \((0, 3)\). This means every part of the parabola is equidistant from this horizontal level line.

The **focus** of the parabola, situated at \((-3, 3)\), plays an equally important role. The parabola is always oriented towards the focus, meaning every point on the parabola is equidistant to the focus and a directrix, creating that iconic "U" shape. Understanding where the focus lies in relation to the vertex helps us determine which direction the parabola opens, leading us to correctly form the parabola equation.

Direction of parabola

The direction a parabola opens is dictated by the geometric relationship between the vertex and the focus. For this exercise, the vertex at \((0,3)\) and the focus at \((-3,3)\) reveal the key detail that this parabola is horizontally oriented.

• If the focus is left of the vertex on the same horizontal line, the parabola opens to the left.
• If the focus were right of the vertex, the parabola would open to the right.

Thus, when given the horizontal positioning of the focus to the left of the vertex, such as in this example, this confirms that the parabola opens leftwards. Knowing this direction is essential to selecting or writing the given parabola's equation in proper form: \(x = A(y-k)^2 + h\) where \((h, k)\) represents the vertex. The direction ultimately affects the sign of the coefficient \(A\), impacting the parabola's size and orientation.

Distance to focus

The distance from the vertex to the focus is another critical feature in defining a parabola. This distance directly influences the parameter \(A\) in the equation \(x = A(y-k)^2 + h\). Specifically, the distance between the vertex and focus equals \(\frac{1}{4A}\). It tells us how "stretched" or "compressed" the parabola will appear.

• In this exercise, the distance from the vertex \((0,3)\) to the focus \((-3,3)\) is precisely 3 units horizontally.
• Using the distance formula, this length can directly allow us to solve for \(A\) in the equation \(1/4A = 3\), leading us to find that \(A = -\frac{1}{12}\).

This calculation ensures the parabola is correctly sized and oriented, finalizing the derived equation as \(x = -\frac{1}{12}(y-3)^2\).