Problem 39 - 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. $$f(x)=3 x^{2}+6 x-4$$

Step-by-Step Solution

Answer

The given quadratic function can be written in the form \(f(x)=3(x+1)^2 - 7\), after completing the square. The vertex of the parabola is (-1,-7) and it has a minimum point at (-1,-7).

Step 1: Write f(x) in a form that we can complete the square

The given function is \(f(x)=3x^{2}+6x-4\). We factor out the coefficient of \(x^{2}\) in the quadratic and linear term, giving us \(f(x)=3(x^2+2x)-4\).

Step 2: Complete the square on the factored quadratic and linear term

We now need to complete the square for the expression \(x^{2}+2x\). The term that needs to be added to complete the square is \((b/2a)^{2}\), where 'b' is the coefficient of 'x' and 'a' is the coefficient of \(x^{2}\). In our case, a=1 and b=2, so \((b/2a)^{2}=(2/2)^{2}=1^{2}=1\). So, \(x^{2}+2x\) can now be completed in square form as \((x+1)^{2}\), resulting in \(f(x)=3[(x+1)^2 - 1] - 4\).

Step 3: Simplify the expression

Simplify the last expression gives us \(f(x)=3(x+1)^2 - 3 - 4 = 3(x+1)^{2}-7\). Hence the function \(f(x)=3x^{2}+6x-4\) can be written as \(f(x)=3(x+1)^{2}-7\) after completing the square.

Step 4: Find the vertex and determine its type

The quadratic function \(f(x)=3(x+1)^2 - 7\) is in standard form, so the vertex of this parabola is (-1,-7). As the coefficient 'a' (which is the coefficient of \((x-h)^{2}\)) is positive, the parabola opens upwards, and so it has a minimum point at (-1,-7).

Key Concepts

Quadratic Functions

Quadratic functions are at the heart of algebra and are typically written in the standard form, which looks like \(f(x) = ax^2 + bx + c\). These types of functions graph into a shape called a parabola, which can open either upwards or downwards depending on the sign of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, while a negative 'a' causes it to open downwards.

Understanding how to manipulate the standard quadratic equation is essential for solving various algebraic problems. Completing the square is a method used to convert a quadratic function into vertex form, namely \(f(x)=a(x-h)^2+k\), where \((h,k)\) is the vertex of the parabola. This transformation is particularly useful for analyzing the properties of the parabola, such as its vertex and whether it represents a maximum or minimum point of the function.

Vertex of a Parabola

The vertex of a parabola is a significant point where the graph either reaches the highest or lowest value depending on the orientation of the parabola. When the parabola opens upward, the vertex is the lowest point, known as the minimum point; conversely, if the parabola opens downward, the vertex is the highest point, or the maximum point.

When a quadratic function is in vertex form \(f(x)=a(x-h)^2+k\), the vertex is easily found at the coordinates \((h,k)\). In the context of completing the square, once the quadratic equation is converted into this form, identifying the vertex is straightforward, and it provides a clear picture of the graph's apex.

Maximum and Minimum Points

In the realm of quadratic functions, maximum and minimum points are pivotal as they indicate the points where the function reaches its extremum. For any given parabola \(y = ax^2 + bx + c\), if 'a' is positive, the parabola has a minimum point, and we can find this minimum at the vertex of the parabola. This point corresponds to the lowest y-value on the graph. Conversely, if 'a' is negative, the parabola has a maximum point at the vertex, correlating to the highest y-value on the graph.

By completing the square and rewriting the quadratic function in the form of \(f(x)=a(x-h)^2+k\), we're better equipped to determine these defining points. Additionally, this form makes the task of differentiating between maximum and minimum points effortless, as it hinges on the sign of 'a'.