Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the parabola with vertex at the origin and axis along the \(x\) -axis if the parabola passes through the point \((3,-1)\). Make a sketch.

Step-by-Step Solution

Answer

The equation of the parabola is \( y^2 = \frac{1}{3}x \).

Step 1: Understanding Parabola's General Form

A parabola with its axis along the positive or negative x-axis and vertex at the origin has the equation of the form \( y^2 = 4px \), where \( p \) is the distance from the vertex to the focus. Since the parabola passes through \((3, -1)\), this step helps us to identify that the equation must represent a sideways opening parabola.

Step 2: Plugging in the Known Point

We know that the point \((3, -1)\) lies on the parabola. Substitute \(x = 3\) and \(y = -1\) into the general equation \( y^2 = 4px \) to solve for \( p \). This gives us: \((-1)^2 = 4p(3)\). Simplifying, we have \(1 = 12p\).

Step 3: Solving for Parameter \( p \)

To find \( p \), solve the equation \(1 = 12p\). Divide both sides by 12 to isolate \( p \), resulting in \( p = \frac{1}{12} \).

Step 4: Writing the Equation of the Parabola

Now that we have \( p = \frac{1}{12} \), substitute \( p \) back into the general equation \( y^2 = 4px \). This gives us \( y^2 = 4\left(\frac{1}{12}\right)x \) or simplified, \( y^2 = \frac{1}{3}x \).

Step 5: Sketching the Parabola

The sketch for \( y^2 = \frac{1}{3}x \) indicates a parabola that opens to the right with a vertex at the origin (0,0), and passing through (3,-1). Since \( p = \frac{1}{12} \), it has a relatively flat opening.

Key Concepts

Vertex

The vertex of a parabola is the point where it changes direction, serving as the parabola's 'cornerstone.' In this exercise, the vertex is at the origin \(0, 0\). When a parabola has its vertex at this spot, it's symmetric about the axes.
In vertex form, the equation of a parabola is \(y = a(x-h)^2 + k\), where \(h\) and \(k\) are the vertex coordinates. However, since the vertex is the origin, the equation simplifies significantly. Hence, the parabola can simply be written in the form of \(y^2 = 4px\), assuming the focus is horizontally placed on the x-axis.
When the vertex is at the origin:

• It becomes the highest or lowest point of the parabola depending on orientation.
• All attributes, such as direction and width, start from this point.

Axis of Symmetry

The axis of symmetry is a vertical or horizontal line that divides the parabola into two mirror-image halves. For our particular problem, the axis is horizontal, parallel to the x-axis, because the parabola equation is \(y^2 = 4px\).
This axis of symmetry is key for understanding how the parabola is oriented. A parabola with a horizontal axis of symmetry is said to "open sideways," either to the left or to the right:

• If \(p > 0\), the parabola opens to the right, as in our case.
• If \(p < 0\), it opens to the left.

Recognizing the axis of symmetry can help predict a parabola's mirror-like behavior and simplify solving problems related to its curve.

Focus of a Parabola

The focus of a parabola is a critical point that defines the curve's shape along with the directrix, a line parallel to the axis of symmetry. In our example, the parabola has the equation \(y^2 = \frac{1}{3}x\), meaning it opens sideways.
The focus lies on the axis of symmetry, a distance \(p\) away from the vertex. We previously calculated \(p = \frac{1}{12}\), indicating the focus is at \(\left(\frac{1}{12}, 0\right)\) along the x-axis from the origin.
Here’s what you need to know about the focus:

• Every point on the parabola is equidistant to the focus and a directrix line that is also \(p\) units from the vertex, but in the opposite direction.
• A smaller value of \(p\) means the parabola will appear "flatter," as it nears the directrix faster.

Understanding the focus helps visualize how the parabola bends and stretches around its central axis.