Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the parabola whose vertex is the origin and whose axis is the \(y\) -axis if the parabola passes through the point \((-3,5)\). Make a sketch.

Step-by-Step Solution

Answer

The parabola's equation is \(x^2 = \frac{9}{5}y\).

Step 1: Understanding the Parabola Equation

A parabola with its vertex at the origin and its axis parallel to the y-axis is represented by the equation \(x^2 = 4py\), where \(p\) is the distance from the vertex to the focus. In this form, all points (\(x, y\)) satisfying \(x^2 = 4py\) lie on the parabola.

Step 2: Substituting the Given Point

We know that the parabola passes through the point \((-3, 5)\). To find \(p\), we substitute \(x = -3\) and \(y = 5\) into the equation \(x^2 = 4py\). This gives us: \[(-3)^2 = 4p(5)\] which simplifies to: \[9 = 20p\].

Step 3: Solving for \(p\)

To solve for \(p\), we need to isolate \(p\) on one side of the equation obtained: \[9 = 20p\] Divide both sides by 20: \[p = \frac{9}{20}\].

Step 4: Formulating the Equation

Now that we have \(p\), we can substitute it back into the general equation of the parabola: \[x^2 = 4py\].This gives us: \[x^2 = 4 \times \frac{9}{20}y\] Simplifying, we get the equation of the parabola: \[x^2 = \frac{36}{20}y\] or further reduced to: \[x^2 = \frac{9}{5}y\].

Step 5: Sketching the Parabola

Since the vertex is at the origin and the parabola passes through (-3,5), it opens upward. Draw the y-axis and x-axis, plot the origin (vertex), and sketch a symmetrical curve passing through the point (-3,5). Ensure it is a parabola that opens upward, with the vertex at (0,0).

Key Concepts

Parabola Equation

A parabola is a symmetrical open plane curve, which is an important part of conic sections. The general form of a parabola's equation with its vertex at the origin and axis parallel to the y-axis is given by:

• \(x^2 = 4py\)

The point known as the vertex of the parabola is where the curve changes direction. With this equation, 'p' plays a crucial role as it represents the distance from the vertex to the focus. When you have the parabola opening upwards or downwards, this formula provides a neat and concise description of how the parabola stretches or compresses. Additionally, every point \((x, y)\) on the parabola satisfies this equation, meaning if you plug in a specific \(x\) and \(y\), they will adhere to this mathematical relationship.

Focus and Vertex

The focus and vertex are essential features of any parabola. For a parabola with a vertical axis, the vertex is typically represented as the "peak" of the parabola. In simplified terms, the vertex is the point at which the parabola makes its sharpest turn.
The focus, on the other hand, is a fixed point used to define the parabola more precisely. In the equation \(x^2 = 4py\), 'p' is critical because it dictates the position of the focus relative to the vertex. When the vertex is positioned at the origin \((0,0)\), and if 'p' is positive, the focus is located at \((0, p)\), above the vertex. Conversely, if 'p' is negative, the focus is \((0, -p)\), below the vertex.
Understanding the location of these important points helps you to accurately sketch and interpret the opening direction and the shape of the parabola.

Solving Algebraic Equations

When solving algebraic equations, especially in the context of finding the parabola equation, substitution and simplification are your best friends. Let’s say we have identified a point that the parabola passes through, for instance

• \((-3, 5)\)

You substitute these coordinates into the parabola's general equation \(x^2 = 4py\) to solve for 'p'.
Start by placing \(-3\) into the equation for \(x\) and \(5\) for \(y\), transforming the equation into:

• \((-3)^2 = 4p(5)\)
• \(9 = 20p\)

Next, to find 'p', divide both sides by 20:

• \(p = \frac{9}{20}\)

This process is essential because each step of substitution and careful simplification brings you closer to the final form of the equation. In this example, once \(p\) was determined, it was plugged back into \(x^2 = 4py\) to provide the specific equation for our parabola:

• \(x^2 = \frac{9}{5}y\)