Problem 50 - College Algebra

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 function is \(y = 2x^2\).

Step 1: Identify the Vertex

The vertex of a quadratic function is the point where the graph changes direction, exhibiting either the highest or lowest value. Inspecting the table, the vertex appears to be at the point where \(x = 0\) and \(y = 0\). Hence, the vertex is \((0, 0)\).

Step 2: Determine the Axis of Symmetry

The axis of symmetry of a quadratic function in the form \(y = ax^2 + bx + c\) is always a vertical line that passes through the vertex. Because the vertex is \((0, 0)\), the axis of symmetry is \(x = 0\).

Step 3: Find the General Form of the Function

Using the vertex form of a quadratic equation \(y = a(x - h)^2 + k\) where \((h, k)\) is the vertex, substitute \(h = 0\) and \(k = 0\) yielding \(y = ax^2\). Now use other points to determine \(a\). Substitute \(x = 1\) and \(y = 2\) in \(y = ax^2\) to get \(2 = a(1)^2\), thus \(a = 2\). Hence the equation is \(y = 2x^2\).

Step 4: Verify the Equation

Verify the derived equation \(y = 2x^2\) with other points: when \(x = -2\), \(y = 2(-2)^2 = 8\); when \(x = 2\), \(y = 2(2)^2 = 8\). The values fit the original table, confirming the equation is accurate.

Key Concepts

Vertex

In a quadratic function, the vertex represents the point where the parabola changes direction. This can be seen as the peak or the lowest point of the curve, depending on whether the parabola opens upwards or downwards.
When examining a set of data points, identifying the vertex is crucial, as it helps to define the parabola's shape.
The vertex can be found where the quadratic reaches its minimum or maximum value in the y-axis. In our given problem, looking at the table, we identify the vertex as the point where both x and y are zero: \(x = 0\) and \(y = 0\).
This means the vertex of the quadratic function is \(\( (0, 0) \)\).
The vertex allows us to rewrite the quadratic equation in a form that makes it clear how the parabola is positioned on the graph.
In general, for quadratic functions, this is described by the vertex form equation: \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex.

Axis of Symmetry

The axis of symmetry is a vital linear feature of any quadratic function's graph.
It is an imaginary line that perfectly splits the parabola into two symmetrical halves. This means that if you were to fold the graph along this line, the two halves would match perfectly.
The axis of symmetry passes through the vertex of the parabola.
In our exercise, knowing the vertex at \(x = 0\), we establish the axis of symmetry as the vertical line \(x = 0\).
This straight line helps us because it confirms the consistent nature of a quadratic curve, guaranteeing that for every point on the left side of the line, there's a mirror image point on the right.
Overall, the equation for the axis of symmetry comes from the formula \(x = h\) where \(h\) is the x-coordinate of the vertex.

Quadratic Equation

The quadratic equation is a second-degree polynomial typically expressed in the standard form: \(y = ax^2 + bx + c\).
This formula describes a parabola on a graph and can be manipulated into different forms for various purposes.
One crucial form is the vertex form, which highlights the vertex, written as \(y = a(x - h)^2 + k\).
Using the vertex \( (0, 0) \) from our exercise, we start shaping our quadratic equation to \(y = a(x - 0)^2 + 0\), simplifying to \(y = ax^2\).
To find the coefficient \(a\), we use another point from our data table. Substituting \(x = 1\) and \(y = 2\) into our equation gives \(2 = a(1)^2\).
Simplifying, we find \(a = 2\). Thus, the quadratic equation perfects to \(y = 2x^2\). This equation can then be verified using other data points to ensure it matches the original values provided.