Problem 66 - College Algebra with Modeling and Visualization

Question

Find an equation of a parabola that satisfies the given conditions. Horizontal axis, vertex \((-1,2),\) passing through \((2,3)\)

Step-by-Step Solution

Answer

The equation is \((y - 2)^2 = \frac{1}{3}(x + 1)\).

Step 1: Identify the Equation Form

For a parabola with a horizontal axis, the standard form is expressed as \[(y - k)^2 = 4p(x - h)\] where \((h, k)\) is the vertex of the parabola. In this problem, the vertex is given as \((-1, 2)\).

Step 2: Substitute the Vertex Coordinates

Insert the vertex coordinates \((-1, 2)\) into the standard form:\[(y - 2)^2 = 4p(x + 1)\].

Step 3: Use Point to Solve for 'p'

The parabola passes through the point \((2, 3)\). Substitute \(x = 2\) and \(y = 3\) into the equation to find \(p\):\[(3 - 2)^2 = 4p(2 + 1)\]\[1 = 12p\]Solving for \(p\), you get \(p = \frac{1}{12}\).

Step 4: Write the Equation

Plug the value of \(p\) back into the equation:\[(y - 2)^2 = \frac{4}{12}(x + 1)\]Simplifying, the equation of the parabola is:\[(y - 2)^2 = \frac{1}{3}(x + 1)\].

Key Concepts

Vertex Form

Understanding the vertex form of a parabola is crucial when dealing with quadratic equations, especially when the parabola's orientation changes. This form is different based on the axis of symmetry it follows. For a parabola with a horizontal axis of symmetry, the vertex form of the equation is given as:

• \[(y - k)^2 = 4p(x - h)\]

In this equation:

• Vertex \( (h, k) \): This point indicates the peak or lowest point of the parabola, depending on its orientation.
• Focus Parameter \( p \): This determines the distance between the vertex and the focus, directly affecting the parabola's width.

The equation reflects the parabola's shape and orientation number through the parameters defined. Notably, a parabola will open either to the left or right if it has a horizontal axis, contrasting with the vertical direction.

Horizontal Axis

The horizontal axis defines a unique orientation of a parabola, which alters the typical "up" or "down" opening commonly seen in most introductory parabola discussions. When a parabola has a horizontal axis of symmetry, it opens to the right or left depending on the position of the focus relative to the vertex. This means the x-values affect the way the parabola stretches or compresses.

• In a horizontal orientation, the positions of the x and y terms in the formula are switched from the typical vertical forms you might know.
• Imagine the U-shape of a vertical parabola, but lying on its side. It's distinctive, as the y-values now determine the vertex horizontally.
• This kind of parabola is characterized by having its squared term as \( (y - k)^2 \), which signifies y-values being pivotal in stretching the shape horizontally.

It's essential to visualize the differences in how parabolas open and orient themselves concerning their axes to grasp transformations fully.

Coordinate Geometry

Coordinate geometry is a powerful tool in understanding the properties and equations of shapes like parabolas in a given plane. By using coordinates, you can precisely define the shape, orientation, and size of geometric figures, essential for solving problems in algebra and calculus.

• Point-Slope Relationships: The coordinates of the vertex and other points (like those on the parabola) help in forming equations that reflect real-world shapes.
• Substitution Method: By substituting values into the equation, such as finding \(p\) when given an extra point, you solidify the parabola's equation with precision.
• Visualizing Movement: Understanding coordinates means you can move shapes around a graph seamlessly without losing their structural integrity, which is vital for solving geometric transformations.

Knowing coordinate geometry allows you to manipulate and interpret the graph of a parabola better, offering insights into algebraic properties and graphical representations.