Problem 37 - Precalculus : Building Concepts and Connections

Question

Write each quadratic function in the form \(f(x)=a(x-h)^{2}+k\) by completing the square. Also find the vertex of the associated parabola and determine whether it is a maximum or minimum point. $$h(x)=x^{2}+x-3$$

Step-by-Step Solution

Answer

The vertex form of the function is \(h(x) = (x+\frac{1}{2})^{2} -\frac{13}{4}\). The vertex of the parabola is \((-1/2, -13/4)\) and this vertex represents a minimum point.

Step 1: Rewrite the quadratic function in the form \(x^{2}+bx+c\)

The function is, \(h(x)=x^{2}+x-3\). So, the quadratic terms are in perfect order.

Step 2: Complete the square

To complete the square, we take half the coefficient of \(x\), square it and then add it to and subtract it from the equation simultaneously. As such, the function becomes: \(h(x) = (x^{2}+x+(\frac{1}{2})^{2})-((\frac{1}{2})^{2}) - 3\). Simplifying this gives: \(h(x) = (x+\frac{1}{2})^{2} -(\frac{1}{4}) - 3\).

Step 3: Convert to vertex form

The simplified version of the function is already in vertex form, namely: \(h(x) = (x+\frac{1}{2})^{2} -\frac{13}{4}\).

Step 4: Identify the vertex

The vertex of function \(h(x)\) is given by the values \((h, k)\), which represents the constants in the equation \(h(x)=a(x-h)^2+k\). In our function \(h(x) = (x+\frac{1}{2})^{2} -\frac{13}{4}\), \(h=-\frac{1}{2}\) and \(k=-\frac{13}{4}\), so the vertex of this parabola is \((-1/2 , -13/4)\).

Step 5: Determine if the vertex is a maximum or minimum

Since the coefficient of \(x^2\) in the original function is positive (1 in this case), our parabola opens upwards and hence the vertex \((-1/2 , -13/4)\) represents a minimum point.

Key Concepts

Vertex Form

The vertex form of a quadratic function is a special way to express the function that helps us easily identify key features, like the vertex of the parabola. The vertex form looks like this: \[ f(x) = a(x-h)^2 + k \] Here,

• \(a\) determines the direction and stretch of the parabola.
• \(h\) and \(k\) represent the coordinates of the vertex \((h, k)\).

By completing the square, we can convert any quadratic equation into this vertex form. When we converted \(h(x)=x^{2}+x-3\), we found that it transformed into \(h(x) = (x+\frac{1}{2})^{2}-\frac{13}{4}\). This gives us the vertex at \((-\frac{1}{2}, -\frac{13}{4})\). Writing the quadratic in vertex form simplifies the process of analyzing the position and the properties of the vertex, making it much easier to graph and understand the behavior of the parabola.

Quadratic Functions

Quadratic functions form the backbone of many mathematical models and applications. They are typically expressed in the form: \[ f(x) = ax^2 + bx + c \] where

• \(a\), \(b\), and \(c\) are constants,
• \(x\) is the variable,
• the term \(ax^2\) is what makes it quadratic.

The quadratic function \(h(x) = x^2 + x - 3\) can be manipulated through various methods like factoring, completing the square, or using the quadratic formula. By completing the square, we've rewritten \(h(x)\) as \((x+\frac{1}{2})^{2} - \frac{13}{4}\), easily revealing the vertex. Hence, understanding quadratic functions is crucial for solving and graphing these parabolic equations.

Parabola Properties

Parabolas are U-shaped graphs produced by quadratic functions. When examining a parabola, its properties reveal a lot about the equation and its graph: - **Direction**: Given by the sign of \(a\) in the quadratic equation. A positive \(a\) opens upwards, and a negative \(a\) opens downwards.- **Vertex**: The highest or lowest point on the parabola, found easily when the function is in vertex form.- **Axis of Symmetry**: The vertical line that passes through the vertex divides the parabola into two mirror images. It can be expressed as \(x = h\).- **Stretch and Compression**: The value of \(a\) also affects how wide or narrow the parabola is.For the function \(h(x) = x^2 + x - 3\), completing the square showed that the parabola opens upwards since \(a = 1\) (positive), with the vertex at \((-\frac{1}{2}, -\frac{13}{4})\). Hence, this point represents the minimum value of the function.