Problem 98 - College Algebra

Question

Write the equation of each parabola in standard form. Vertex: \((-3,-1) ;\) The graph passes through the point \((-2,-3)\)

Step-by-Step Solution

Answer

The standard form of the given parabola is \(y=-2(x+3)^2-1\).

Step 1: Insert the vertex into the standard equation

Replace \(h\) and \(k\) with the coordinates of the vertex \((-3,-1)\) so that we obtain \(y=a(x+3)^2-1\).

Step 2: Determine 'a' using the given point

We know that the graph passes through the point \((-2,-3)\), these coordinates will be substituted in \(x\) and \(y\) in our equation to solve for \(a\). Now we get \(-3=a(-2+3)^2-1\) , which simplifies to \(-3=a-1\). Therefore, \(a=-2\).

Step 3: Write the standard form of the parabola

Replace \(a\) in the equation obtained in step 1 with the result from step 2 to obtain the final equation: \(y=-2(x+3)^2-1\).

Key Concepts

Vertex Form of a Parabola

Parabolas, the U-shaped graphs, have a special form called the vertex form. This form highlights the most critical point of the parabola—its vertex. The vertex is essentially the peak of the parabola if it opens downwards, or the lowest point if it opens upwards. To write a parabola in vertex form, use the formula:

• \( y = a(x-h)^2 + k \)

Here, \((h, k)\) represents the vertex of the parabola. The value of \(a\), the "stretch factor," determines whether the parabola opens upwards or downwards, and how "wide" it appears. A positive \(a\) value indicates the parabola opens upward, while a negative \(a\) suggests it opens downward.
Using this form helps with graphing because, unlike the standard form, the vertex form immediately tells us the exact coordinates of the vertex. Understanding how to manipulate the vertex form to reveal properties of the parabola can greatly aid in solving geometry problems.

Parabola Equation

A parabola is defined mathematically by its equation. The equation of a parabola describes the set of points that make up the curve. In geometry, there are multiple ways to represent parabolas. One popular form is the standard form which can be difficult for identifying the vertex easily. However, the vertex form \( y = a(x-h)^2 + k \) is particularly useful as it directly indicates the vertex. This equation simplifies a lot of calculations when you’re handling vertex-related questions.
The given vertex form uses transformations of the basic parabola \( y = x^2 \). By shifting up or down, left or right, you can effectively "move" this parabola into new positions. The vertex form equation also allows you to scale or reflect the parabola, giving you comprehensive control over your graph. Understanding these transformations and how to calculate them forms the foundation for breaking down and analyzing complex quadratic functions.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to solve geometric problems using numerical coordinates and algebra. It connects algebraic equations with geometric figures, making it a powerful tool in mathematics. By using coordinates, you can describe geometric shapes analytically and figure out various properties.
For parabolas, coordinate geometry helps you find important details such as the vertex, axis of symmetry, and the direction in which the parabola opens. By leveraging coordinates, you can plug values into parabola equations and visually examine shifts along the coordinate plane.

• Find specific points: Using geometry, you can locate any point on a parabola via its equation.
• Navigation: Coordinate geometry allows you to easily plot and transform parabolas within the coordinate plane.

This approach provides a systematic way for visualizing and solving real-world problems by modeling them with parabolic shapes.