Problem 12 - College Algebra

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$f(x)--2(x+4)^{2}-8$$

Step-by-Step Solution

Answer

The vertex of the given quadratic function is (-4, -8).

Step 1: Identifying the Standard Form of Quadratic Function

Given function is \(f(x) =-2(x + 4)^2 - 8\). The standard form of a quadratic function is \(f(x) = a(x - h)^2 + k\), where 'a' is the coefficient of the square term, 'h' is the x-coordinate of the vertex and 'k' is the y-coordinate of the vertex.

Step 2: Comparing the Given Function with Standard Form

Comparing the given function with the standard form gives: \(a = -2, h = -4, k = -8\). Hence, the coordinates of the vertex of the given function are (-4, -8).

Step 3: Conclusion

Accordingly, the coordinates of the vertex for the parabola defined by the given quadratic function \(f(x) = -2(x + 4)^2 - 8\) are (-4, -8).

Key Concepts

Parabola

A parabola is a U-shaped curve that you often see in mathematics when dealing with quadratic functions. It can open upwards or downwards based on a particular part of the equation, which we will cover later. The shape of the parabola is symmetrical, meaning if you fold it along its vertical line of symmetry, both halves match perfectly.
The direction in which the parabola opens is primarily determined by the coefficient of the square term in the quadratic function. If this coefficient is positive, the parabola opens upwards like a regular U. If negative, it opens downwards, forming an upside-down U or an "n" shape. For example, in the function in our exercise, the negative coefficient \(-2\) tells us that the parabola opens downwards.

• Symmetry: Parabolas have an axis of symmetry, which is a vertical line that divides the parabola into two equal halves.
• Direction: The leading coefficient determines if the parabola opens up or down.

Recognizing these characteristics can help you in graphing the parabola and understanding the nature of the quadratic function you are working with.

Vertex

The vertex of a parabola is a crucial point. It is where the parabola turns and is either the lowest or highest point on the graph, depending on the parabola's direction. In standard form, the vertex can be easily identified using the values in the equation.To find the vertex in the standard form of a quadratic function \(f(x) = a(x-h)^2 + k\), you use \(h\) and \(k\) in the equation as the coordinates for the vertex, which are \((h, k)\). This means that if you know \(h\) and \(k\), you can instantly locate the vertex on the graph.
In the provided exercise:

• Your vertex is \((-4, -8)\).
• It represents the highest point on the graph because the parabola opens downward.

Understanding the vertex helps you describe the parabola's shape, position on the graph, and also how it shifts in relation to the basic parabola \(x^2\).

Standard Form of Quadratic Function

The standard form of a quadratic function is written as \(f(x) = a(x-h)^2 + k\). This form is helpful because it immediately gives you insight into the parabola's characteristics. You can identify the vertex, the direction in which the parabola opens, and also understand transformations from the basic \(x^2\) graph.The key components in this form include:

• \(a\): The coefficient affecting the vertical stretch or compression of the parabola. If \(|a|>1\), the parabola is narrower, and if \(0<|a|<1\), the parabola is wider.
• \(h\): The horizontal shift from the origin. It's part of the vertex coordinates.
• \(k\): The vertical shift, which is also the y-coordinate of the vertex.

In the exercise, you compared the function \(f(x) = -2(x+4)^2 - 8\) with the standard form to easily find that \(h = -4\) and \(k = -8\), defining the vertex. This form is a powerful tool because it simplifies the process of analyzing quadratic functions.