Problem 40 - College Algebra with Modeling and Visualization

Question

Find an equation of the parabola with vertex \((0,0)\) that satisfies the given conditions. Vertical axis, passing through \((-2,3)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{3}{4}x^2 \).

Step 1: Understand the Parabola's Equation with a Vertical Axis

A parabola with a vertical axis and the vertex at the origin has the general equation \( y = ax^2 \). Here \(a\) is a constant that determines the parabola's openness and direction.

Step 2: Substitute the Given Point into the Equation

To find \(a\), we substitute the coordinates of the point \((-2, 3)\) into the equation. This gives us: \(3 = a(-2)^2\).

Step 3: Solve for the Constant \(a\)

In the equation \(3 = a(-2)^2\), simplify the right-hand side to obtain \(3 = a \cdot 4\). Solve for \(a\) which gives \(a = \frac{3}{4}\).

Step 4: Write the Equation of the Parabola

Using the value of \(a\) obtained, substitute back into the general equation to get \( y = \frac{3}{4}x^2 \).

Key Concepts

Vertex Form

When dealing with parabolas, the vertex form is a handy way to express the equation of a parabola. It's especially useful because it gives you information about the parabola's vertex directly from the equation. The vertex form of a parabola's equation is written as:

• \( y = a(x-h)^2 + k \)

Here, \(h, k\) is the vertex of the parabola. The variable \(a\) affects how "open" the parabola looks and which direction it opens.
You can see that if the vertex is at the origin \(0,0\), the vertex form simplifies to \( y = ax^2 \). This simpler form is used when the parabola's vertex is at the origin, as in our exercise.
What's important is that the value of \(a\) determines if the parabola opens upwards (if \(a > 0\)) or downwards (if \(a < 0\)). Understanding the vertex form makes it easier to visualize and graph the parabola, providing a clear idea of where it is positioned on the graph.

Vertical Axis

Parabolas can have either a vertical or horizontal axis. In this exercise, we focus on a vertical axis. A parabola with a vertical axis is one that opens up or down rather than side to side.
The equation for such a parabola is \( y = ax^2 \), which means it is symmetric about the y-axis, making it straightforward to sketch.
Visualizing the parabola with a vertical axis helps reveal its symmetrical nature, mainly when using points around the vertex. In our given problem, since the parabola passes through the point \((-2, 3)\), this informs us of how the curve shifts and opens around the vertex at \(0,0\).
Considering the axis orientation is crucial as it helps predict how the parabola behaves and interacts with other points and lines on a graph, simplifying problem-solving in many cases.

Constant Determination

Once you've decided the form and axis of your parabola, determining the constant \(a\) is the next critical step. This constant is vital because it dictates the openness of the parabola.
To find \(a\), you substitute a known point through which the parabola passes into your equation. In the example, this point is \((-2, 3)\).

• Substitute \(x = -2\) and \(y = 3\) into the general equation: \(3 = a(-2)^2\).
• Simplify: \(3 = a \cdot 4\).
• Solve for \(a\): \(a = \frac{3}{4}\).

By determining \(a\), you can write the complete equation of the parabola. For our exercise, it results in the equation \( y = \frac{3}{4}x^2 \). Understanding how this affects the shape can guide further analysis or application of parabolas in practical scenarios.