Problem 31 - 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 \((-2, 5)\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(y = 5/4 \cdot x^2\).

Step 1: Interpretation of the problem

We need to find the equation of a parabola with a horizontal axis and vertex at the origin. We know that a point on this parabola is (-2, 5). The standard equation for a parabola with such characteristics is in the form: \(y=a \cdot x^2\) where a is a real constant.

Step 2: Find the value of the coefficient 'a'

We can find the value of 'a' by substituting the known point values into the parabola equation. So, for \(x = -2\) and \(y = 5\), the equation becomes: \(5 = a \cdot (-2)^2\). This simplifies to \(5 = 4a\). Then by dividing both sides of the equation by 4, we find that \(a = 5/4\).

Step 3: Formulate the final equation

Now that we have the value of 'a', we can substitute it back into the original equation. Hence, the standard form of the equation of the parabola is \(y = 5/4 \cdot x^2\).

Key Concepts

Parabola Standard Form

Understanding the standard form of a parabola equation is crucial for analyzing its properties and graphing it correctly. In its most basic form, the equation of a parabola with the vertex at the origin can be written as either \(y=ax^2\) for parabolas that open upwards or downwards, or \(x=ay^2\) for those that open to the right or left.

The coefficient \(a\) in these equations determines the \

Finding Parabola Coefficient

Identifying the coefficient \(a\) in the parabola's equation is key to understanding the parabola's shape and direction. This coefficient affects the parabola's width, direction, and whether it opens up or down (for vertical parabolas) or right or left (for horizontal parabolas).

To find the coefficient \(a\), we use a known point on the parabola. We substitute the x and y values from this point into the parabolic equation and solve for \(a\). For example, if we have a point on the parabola \(y=ax^2\) and that point is \(\(x_0\), \(y_0\))\), we would substitute these values to get \(\(y_0\) = a\(x_0^2\))\). Solving this equation for \(a\) gives us the necessary coefficient to describe our parabola fully.

Vertex at the Origin

In the context of parabolas, the vertex represents the highest or lowest point of the curve, depending on the orientation of the parabola. When a parabola has its vertex at the origin, its equation simplifies significantly.

The vertex form of a parabola's equation is generally \(y=a(x-h)^2+k\), where \(\(h,k\))\) is the vertex. If the vertex is at the origin, this means \(h=0\) and \(k=0\), simplifying the vertex form to \(y=ax^2\) or \(x=ay^2\), depending on its axis of symmetry. This simplified form makes it easier to manipulate and understand the properties of the parabola.