Problem 28 - Precalculus

Question

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

The standard form of the equation of the parabola with a horizontal axis, passing through the point (3,-2) and with the vertex at the origin is \( y^2 = -4/3x \).

Step 1: Determine the orientation of the parabola

The orientation of the parabola will be determined by the characteristic that the axis is horizontal. If the axis is horizontal and we have a point whose y-coordinate is negative, then the parabola opens to the left. Therefore, the standard form of the equation of the parabola with a horizontal axis and vertex at the origin is given by \( y^2 = 4ax \). Because the parabola opens to the left, a will be negative

Step 2: Substituting known values into the equation

We're given a point (3,-2) through which the parabola passes. Substituting x=3 and y=-2 in the standard form of the equation we get \( (-2)^2=4a(3)\)

Step 3: Solve for a

Solving the equation from step 2, we get \( 4=12a \) which simplifies to \( a=\frac{4}{12}=\frac{1}{3} \). However, since the parabola opens to the left, a will be negative. Hence, a = -1/3.

Step 4: Write the standard form equation of parabola

Substituting the value of a back into the standard form of the equation of a parabola gives the equation: \( y^2 = 4*(-1/3)*x \), which simplifies to \( y^2 = -4/3x \).

Key Concepts

Parabola Orientation

Understanding the orientation of a parabola is crucial when you're trying to graph it or write its equation in standard form. The orientation refers to the direction the parabola opens up on a Cartesian plane. There are four possible orientations: upwards, downwards, to the left, or to the right.

For a parabola with its vertex at the origin, the general equation is either in the form of \( y^2 = 4ax \), if it opens horizontally (left or right), or \( x^2 = 4ay \), if it opens vertically (up or down). The variable 'a' in these equations plays a determining role. If 'a' is positive, the parabola opens to the right or upwards; and if 'a' is negative, it opens to the left or downwards.

In our exercise, the given point (3, -2) lies on the parabola, and the parabola has a horizontal axis. Because the y-coordinate of the point is negative, and the parabola's orientation is horizontal, this tells us that our parabola opens to the left. Thus, we'll use the form \( y^2 = 4ax \) with a negative 'a'.

Vertex of a Parabola

The vertex of a parabola is a significant point, representing the peak or the lowest point of the curve, depending on its orientation. It's the point where the parabola changes direction. In other terms, it's the maximum or the minimum point of the parabola. The location of the vertex provides important information for writing the equation of the parabola in standard form.

For a parabola with the standard form of \( y^2 = 4ax \) or \( x^2 = 4ay \), when the vertex is at the origin (0,0), it simplifies the equation since there’s no need to shift the graph horizontally or vertically. In the problem given, since the vertex is at the origin, our equation will not have additional terms for the vertex. This makes plotting and solving much simpler.

Solving Equations

Solving equations is a foundational skill in algebra, which allows us to find unknown values, such as the 'a' parameter in our parabola's equation. To solve equations, we often perform a series of algebraic manipulations, which includes substituting known values, isolating variables, and simplifying expressions.

In the example we’re working with, we substituted the coordinates of the given point into our standard form equation to find the value of 'a'. By squaring the y-coordinate and multiplying by the x-coordinate, we came up with a simple linear equation that we could solve for 'a'. After calculating 'a', we knew that it had to be negative because of the parabola's orientation, so we were able to determine that \( a = -\frac{1}{3} \). This value was then substituted back into the standard form to yield the final equation of our parabola: \( y^2 = -\frac{4}{3}x \).

Remember, when solving equations, every step should bring us closer to isolating the variable we're solving for, and the result must always adhere to the conditions outlined by the rest of the problem.