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 is \( y^2 = \frac{1}{3}x \).

Step 1: Understand the Parabola's Equation

For a parabola with its vertex at the origin and its axis along the x-axis, the general equation is of the form \( y^2 = 4px \). Here, \( p \) is the distance from the vertex to the focus. Since the parabola's axis is along the x-axis, \( y \) values depend on \( x \).

Step 2: Use the Given Point to Find p

The parabola passes through the point \((3, -1)\). Substitute these values into the equation \( y^2 = 4px \). This gives:\( (-1)^2 = 4p(3) \).Simplifying, we get:\( 1 = 12p \).

Step 3: Solve for p

Solve the equation from Step 2 for \( p \):\( p = \frac{1}{12} \).

Step 4: Write the Parabola's Equation

Now that we have found \( p \), substitute it back into the equation \( y^2 = 4px \) to get the final equation:\( y^2 = \frac{1}{3}x \).

Step 5: Sketch the Parabola

To sketch the parabola, note that it opens to the right along the x-axis because the equation is in the form where \( y^2 \) determines \( x \). The vertex is at the origin \((0, 0)\), and it passes through the point \((3, -1)\). The parabola is relatively flat due to the small value of \( p \).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a critical point that often serves as a starting point in understanding its geometry and equation. When dealing with parabolas, the vertex is the point where the curve changes direction. For some, it's seen as the 'tip' of the parabola.In the context of the given problem, the vertex is conveniently located at the origin, (0, 0). This positioning makes it easier to work with the standard form equation of a parabola,

• For parabolas aligned along the x-axis: \(y^2 = 4px\)
• For parabolas aligned along the y-axis: \(x^2 = 4py\)

In our exercise, having the vertex at the origin and the given point (3, -1) was essential in finding the precise equation of the parabola. Remember, the vertex isn't just a point; it's the heart of the parabola where most calculations begin.

Axis of Symmetry

The axis of symmetry in a parabola is a straight line that divides the curve into two identical halves. Imagine it as a mirror line where one side of the parabola reflects onto the other. Understanding this line helps in sketching the parabola accurately and comprehensively.For the parabola in the exercise, the axis of symmetry is along the x-axis. This means the parabola opens horizontally. This characteristic of symmetry means that reflecting the points across the x-axis yields the rest of the parabola.A clear hint that a parabola has its axis along the x-axis in an equation is when the equation is in the form \(y^2 = 4px\). The symmetry of a parabola makes setting it on a graph more predictable. Knowing the axis of symmetry can guide you not only mathematically but visually.

Focus of a Parabola

The focus of a parabola is a special point located inside the curve. It's the point around which the parabola is 'dragged'. Any point on the parabola is equidistant to the focus and a line called the directrix.In this exercise, once you've determined that the equation is \(y^2 = 4px\) and you found \(p = \frac{1}{12}\), you can conclude that the focus lies at the point (\(\frac{1}{12}\), 0). This implies that the focus is very close to the origin, guiding us to the next key property.

• The formula \(4p\) encompasses the geometric properties of the parabola.
• If \(p\) is positive, the parabola opens rightwards; if negative, leftwards.
• The closer the focus, the steeper the parabola's opening.

Knowing the focus assists in understanding the depth and direction of the parabola’s curve. It keeps the relationship between all elements of the parabola coherent and predictable.