Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex: \((-3,5)\); point: \((-6,-1)\)

Step-by-Step Solution

Answer

The equation of the parabola with vertex at (-3,5) and passing through the point (-6,-1) is \(y=-1(x+3)^2+5\).

Step 1: Identify the vertex (h,k)

The vertex of the parabola is given as (-3,5). So, the equation of the parabola can be recorded as \(y=a(x + 3)^2+5\).

Step 2: Compute 'a' value

We will find the 'a' value with the other given point (-6,-1) that is on the parabola. Substitute x=-6 and y=-1 in the equation: -1 = a (-6+3)^2 +5. Simplify this equation to find the value of a which is -1.

Step 3: Form the equation of parabola

Plug the a value from previous step into the parabola equation we have: \(y=-1(x+3)^2+5\).

Key Concepts

Vertex Form

When dealing with parabolas, one method to write the equation is in the vertex form. This particular way of expressing a parabola is great because it gives you a clear view of the parabola's most important point: the vertex. The vertex form of a parabola is written as:

• \( y = a(x-h)^2 + k \)

Here,

• \( (h, k) \) represents the vertex of the parabola. It's the highest or lowest point on the graph, depending on the orientation of the parabola.
• \( a \) is a constant that affects the width and the direction of the parabola. If \( a \) is positive, the parabola opens upwards, and if it's negative, it opens downwards.

Knowing how to manipulate these elements allows you to easily graph the function or solve for additional points that lie on it. In our exercise, we start with the vertex \((-3, 5)\), instantly giving us the framework for our equation.

Coordinate Geometry

Coordinate geometry provides tools for solving geometrical problems by means of algebra. When graphing parabolas, you'll often deal with two points: the vertex and another point on the parabola. These points lie in the coordinate plane and provide crucial information for forming equations.
Using coordinate geometry, we gather:

• Vertex: Provides a point that is central to the parabola's symmetry.
• Additional Point: Verifies or helps determine specific values in the equation, like the coefficient \( a \).

In the example provided, with a given vertex of \((-3,5)\) and another point \((-6,-1)\), we can use these coordinates to find the entire algebraic representation of the parabola through substitution into the vertex form equation. Coordinate geometry tools allow us to translate these physical points into mathematical expressions.

Quadratic Equations

Quadratic equations form the backbone of many mathematical concepts and problems, especially when it comes to parabolas. A quadratic equation typically takes the form of:

• \( ax^2 + bx + c = 0 \)

However, in the vertex form, you can see how it pivots into:

• \( y = a(x-h)^2 + k \)

Both forms describe quadratic relationships, but the vertex form emphasizes positions such as the vertex itself.
The concept of deriving the quadratic form from known points, like in our exercise, helps us stitch together the known characteristics of a parabola. Here:

• "\( a \) value" adjustment: Determined by substituting a known point \((-6, -1)\) into the equation.

This ensures that the graph of the equation actually passes through the provided point. Hence, quadratic equations represent not just a plot on the chart but a relationship between the variables that can be manipulated to achieve various solutions.