Problem 46 - 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. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 5 & 2 & 1 & 2 & 5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The quadratic function is \(y = x^2 + 1\).

Step 1: Identify the Symmetry

Observe the given points. They look symmetric around the y-axis, which suggests the function's axis of symmetry could be at \(x = 0\).

Step 2: Determine the Vertex

Since the values show symmetry around the y-axis and the function is symmetric about \(x = 0\), the point at \((0, 1)\) is likely the vertex of the parabola.

Step 3: Write the Vertex Form

The vertex form of a quadratic equation is given by \(y = a(x-h)^2 + k\), where \((h, k)\) is the vertex. Substituting the vertex \((0, 1)\) into the equation gives us \(y = a(x-0)^2 + 1\).

Step 4: Find the Value of 'a'

Using another point from the table, such as \((-1, 2)\), substitute into the equation \(y = ax^2 + 1\). So, \(2 = a(-1)^2 + 1\). This simplifies to \(2 = a + 1\), thus \(a = 1\).

Step 5: Write the Quadratic Equation

With \(a = 1\), the quadratic function is \(y = (x-0)^2 + 1\), which simplifies to \(y = x^2 + 1\).

Key Concepts

Vertex Form

Quadratic functions can often be expressed in what's called the "vertex form." This form makes it easy to identify key features of the function, especially the vertex. The equation is written as \(y = a(x-h)^2 + k\). Here:

• \(a\) determines the width and direction of the parabola.
• \((h, k)\) represents the vertex.

This form is particularly useful because it visually represents how to shift the graph of \(y = x^2\) horizontally by \(h\) units and vertically by \(k\) units. In the given problem, once we found the vertex to be \((0, 1)\), we immediately used the vertex form as \(y = a(x-0)^2 + 1\). This step simplifies the process of finding the quadratic function by directly connecting the graph's shape and location to the equation.

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that evenly divides the parabola into two mirrored halves. This is an essential feature as it often passes through the vertex, making the vertex the top or bottom point of the parabola, depending on its orientation. For any quadratic in vertex form, the axis of symmetry is always given by \(x = h\), where \(h\) is the x-coordinate of the vertex. In our exercise, with the vertex at \((0, 1)\), the axis of symmetry is the line \(x = 0\). Identifying this axis helps not only in graphing but also in ensuring that the transformation of the parabola preserves its symmetry attribute.

Parabola

A parabola is the graph of a quadratic function. It is a U-shaped curve that can open upwards or downwards. The orientation and width of the parabola are controlled by the coefficient \(a\) in the equation \(y = a(x-h)^2 + k\). If \(a\) is positive, the parabola opens upward, and if \(a\) is negative, it opens downward. The absolute value of \(a\) affects the "width" of the parabola: larger values make it narrower, and smaller values make it wider. In the provided exercise, our calculation showed \(a = 1\), resulting in a parabola that opens upwards, confirming its upward U-shape with points efficiently calculated around the vertex.

Vertex Point

The vertex point is perhaps the most crucial point on the parabola, as it represents the maximum or minimum of the function, depending on its orientation. In quadratic functions, knowing the vertex allows a complete visualization of the function's turning point. Our table of values in the exercise highlighted the vertex at \((0, 1)\). This point marks the lowest vertex of the curve since \(a\) is positive and the parabola opens upwards. Understanding the position of the vertex aids not only in graph plotting but in solving problems related to maximum or minimum values in quadratic contexts.