Problem 32 - Precalculus with Limits

Question

In Exercises 19-32, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Horizontal axis and passes through the point \((3, -2)\)

Step-by-Step Solution

Answer

Therefore, the standard form of the equation of the parabola is \(y^2 = 4/3x\).

Step 1: Substitute the given point into the equation

To find the value of \(a\), the given point \((3, -2)\) is substituted into the standard equation of the parabola. The x-coordinate in the point is the value of x, and the y-coordinate is the square of the y in the equation. This gives us \((-2)^2 = 4a(3)\), which simplifies to \(4 = 12a\).

Step 2: Solve for a

The equation \(4 = 12a\) is then solved for \(a\). This can be done by dividing both sides of the equation by 12, giving \(a = 4/12 = 1/3\).

Step 3: Write the standard form of the equation of the parabola

Now that the value of \(a\) is known, it is substituted back into the standard equation to find the equation of the parabola. This gives us \(y^2 = 4(1/3)x\), which simplifies to \(y^2 = 4/3x\).

Key Concepts

Vertex Form

A key aspect of understanding parabolas is getting to know the vertex form of the equation. The vertex form is particularly useful because it directly includes the coordinates of the parabola's vertex, making it easy to identify. This form looks like this:

• The general vertex form of a parabola equation is: \( y = a(x-h)^2 + k \).
• Here, \( (h, k) \) represents the vertex of the parabola.
• The value of \(a\) determines the direction and "width" of the parabola.

If \( a > 0 \), then the parabola opens upward. Conversely, if \( a < 0 \), it opens downward. The better you understand vertex form, the easier it becomes to sketch parabolas and identify their key features. Starting with the vertex form also makes transitioning to other forms, like the standard form, smoother. Just remember, having the vertex at the origin \((0,0)\) simplifies the equation to \( y = ax^2 \) without the \( h \) and \( k \) terms.

Standard Form

The standard form of a parabola's equation is another common format that provides essential information. It specifically focuses on the relationship between the squared and non-squared terms. A parabola's standard form can be particularly beneficial in identifying whether the parabola has a horizontal or vertical axis. For a parabola with a vertical axis, the standard equation is:

• \( y = ax^2 + bx + c \)

However, when the parabola's axis is horizontal, like in our exercise, the roles of \(x\) and \(y\) are swapped:

• \( y^2 = 4ax \)

The key points to remember include:

• "\(a\)" influences how "wide" the parabola is and whether it opens to the right or left
• For a horizontal parabola, if \(a > 0\), it opens to the right. If \(a < 0\), it opens to the left.
• The constant term "4" in "\(4a\)" is a reflection of the geometry of the curve, indicating that the parabola is sized similarly in each direction.

In our example with a horizontal axis of symmetry and a vertex at the origin, it simplifies to \(y^2 = 4/3x\), which helps in plotting and understanding how the parabola behaves.

Horizontal Axis

When tackling parabolas, one might encounter a horizontal axis of symmetry. This is different from the more common vertical axis found in many quadratic functions.

• The axis of symmetry for a parabola is a line that divides the parabola into two mirror-image halves.
• For parabolas with horizontal axes, this line runs from left to right, as opposed to top-down as seen with vertical axes.
• The horizontal axis results in parabolas where the squared term involves \(y\), such as in \(y^2 = 4ax\).

In the context of our original exercise:

• The horizontal axis is key because it leads to a horizontal "opening" direction (left or right).
• Since our parabola goes through the point \((3, -2)\), we solved for "\(a\)" in the horizontal form, leading to the equation \(y^2 = 4/3x\).

This type of understanding allows one to sketch the curve correctly and comprehend how varying formulae affect the graph's direction and orientation. In summary, mastering the concept of a horizontal axis can demystify a slew of problems related to parabola structures.