Problem 52 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (0,3)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola with focus at (0,3) and vertex at the origin is \(y =(1/12)x^2\)

Step 1: Determine the direction of the parabola

From the coordinates given for the focus (0,3), we can determine that the parabola opens upwards. Therefore, we can use the standard form: \(y = ax^2\). The next step is to determine the value of 'a' to complete the standard equation.

Step 2: Calculate the value of 'a' using the distance formula

The vertex is at the origin and the focus is at (0,3). The distance between them is 3 units which is the value of 'a'. Therefore the 'a' in the standard form of parabolic equation will be 1/(4*3) as parabola standard form is \(y = a(x-h)^2 + k\), where a is 1/(4*f) and f is the distance.

Step 3: Write the equation of the parabola

Using a = 1/12 calculated in step 2, substitute 'a' into the standard form of parabola equation yielding \(y = (1/12)x^2\)

Key Concepts

Standard Form Equation

Understanding the standard form of a parabola's equation is crucial when dealing with its geometry. A parabola that opens upwards or downwards can be represented by the equation:

• \[ y = a(x-h)^2 + k \]

In this formula,

• \( (h, k) \) is the vertex of the parabola, which is a point where the parabola changes direction.
• The variable \( a \) determines the "width" and the "direction" (upward if \( a > 0 \) and downward if \( a < 0 \)) of the parabola.

In the given problem, because the parabola opens upwards and the vertex is at the origin \((0, 0)\), the equation simplifies to \( y = ax^2 \). From here, finding the exact value of \( a \) is essential to tailor the equation to the parabola's specific characteristics.

Focus of a Parabola

The focus is one of the unique characteristics of a parabola and is defined as a fixed point inside the curve. All points on a parabola are equidistant from the focus and the directrix of the parabola. In the exercise, the provided focus is at (0, 3). This focus helps determine the parabola's orientation and the value of \( a \) in its standard form equation.

• For a parabola with its vertex at the origin \((0,0)\) and the focus at \((0,f)\), the formula for \( a \) can be determined as \( a = \frac{1}{4f} \).
• Here, \( f \) is the focal length, the distance between the vertex and the focus.

Thus, knowing the focus allows us to compute this distance, giving us the means to accurately mold the parabola's equation by replacing \( f \) with the non-zero value provided, and ultimately zeroing in on a complete expression.

Vertex of a Parabola

The vertex of a parabola is the point where it turns; it is either the lowest or highest point depending on the parabola's direction. For the problem tackled here, the vertex is at \((0,0)\), pinpointing exactly where the parabola's axis of symmetry lies.

• For parabolas described with vertex \((h, k)\), the symmetry line is \( x = h \), and this key point informs both the shape and position.
• Here, because both \( h \) and \( k \) are zero, the vertex form of the equation simplifies to \( y = ax^2 \), meaning alignment along the y-axis.

The vertex plays an instrumental role, serving as the parameter from which distances to both the focus and directrix are measured. Thus, the exercise uses this point to directly influence the equation structure and determine additional constants.