Problem 50 - College Algebra with Corequisite Support

Question

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 8 & 2 & 0 & 2 & 8 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The quadratic equation is \(y = 2x^2\), with vertex at (0, 0).

Step 1: Identify Symmetrical Points

Observe the table to identify points that are symmetrical relative to each other. The points \((-2, 8)\) and \((2, 8)\) are symmetrical around \(x = 0\); similarly, \((-1, 2)\) and \((1, 2)\) are symmetrical around \(x = 0\). This suggests that the axis of symmetry is \(x=0\).

Step 2: Determine the Vertex

Since the axis of symmetry is \(x = 0\), identify the point with \(x = 0\) from the table. The vertex is thus at \((0, 0)\).

Step 3: General Form of the Quadratic Equation

Recognize that the vertex form of a quadratic equation is \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex. Here, \(h = 0\) and \(k = 0\), making it \(y = ax^2\).

Step 4: Find Coefficient 'a'

Use another point from the table to find \(a\). Take \((1, 2)\) as an example. Substitute into the vertex form: \(2 = a(1)^2\). Solve for \(a\): \(a = 2\).

Step 5: Write the Final Equation

Using the vertex form and the determined \(a\), the equation is \(y = 2x^2\). Verify using other points to confirm correctness.

Key Concepts

Axis of Symmetry

In the realm of quadratic functions, the axis of symmetry is a crucial concept. It is a vertical line that divides the parabola into two mirror-image halves, effectively making the graph symmetrical. In the original exercise, we were given a table of points, and from this, we needed to determine the axis of symmetry.
This task was simplified by identifying symmetrical pairs of points. For instance:

• The points \((-2, 8)\) and \(2, 8\)\ are symmetrical as they are equally distanced from the line \(x = 0\)\.
• Similarly, \((-1, 2)\) and \(1, 2\)\ hold this symmetrical property.

By recognizing these symmetrical points, we can confidently say the axis of symmetry is indeed \(x = 0\)\. This knowledge is fundamental for graphing the function and understanding its behavior. The axis of symmetry provides a framework around which the rest of the quadratic equation is constructed.

Vertex Form

Understanding the vertex form of a quadratic equation is like holding the blueprint to a structure. The vertex form gives us the essential information needed to build the graph of a quadratic function. The form is articulated as \(y = a(x-h)^2 + k\), where \( (h, k) \) represents the vertex of the parabola.
In our problem, after identifying the axis of symmetry, the vertex was pinpointed as \( (0, 0) \). This moment is key because it tells us where our parabola will turn or shift direction. This is the lowest or highest point on our graph depending on the parabola's orientation.

• When \(h = 0\) and \(k = 0\), our equation starts as \(y = a(x-0)^2 + 0\), simplifying to \(y = ax^2\).
• The value \(a\) influences whether the parabola opens upwards (a positive value) or downwards (a negative value) and affects its width.

Thus, utilizing the vertex form not only helps us plot each part of the curve accurately but also allows us to determine the parabola's exact nature easily.

Symmetrical Points

Symmetrical points in a quadratic function are pairs of points that are equidistant from the axis of symmetry. They are a fascinating feature of parabolas since they visually balance the function graph and provide critical insights.
In our exercise's given table of values:

• Points like \((-2, 8)\) paired with \( (2, 8) \) illustrate symmetry beautifully. The y-values are identical, and the x-values are equal distances from zero.
• Similarly, \( (-1, 2)\) and \( (1, 2) \) demonstrate this balance around the axis of symmetry in the same fashion.

Grasping this symmetry helps us confirm the axis of symmetry and gives a greater understanding of how the vertex anchors the entire structure of the graph. These symmetrical features also let us check our work when deriving the quadratic equation. If plotted, the points and their symmetry can visually validate our calculated operations.