Problem 38 - 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. $$g(x)=-x^{2}+x-7$$

Step-by-Step Solution

Answer

The given quadratic function in the form \(g(x)=a(x-h)^2+k\) is \(g(x)=-1(x-0.5)^{2}-6.75\). The vertex of the function is at \((0.5, -6.75)\) and it is a maximum point.

Step 1: Write the function in vertex form

To write the function in vertex form, you first need to complete the square. The formula to complete the square is \((b/2)^2\). But before that, separate the \(x\)-terms from the constants: \(g(x) = -(x^{2} - x) - 7\). Now complete the square: \(g(x) = -(x^{2} - x + 0.25) - 7 + 0.25 = -(x-0.5)^{2} - 6.75\)

Step 2: Identify the vertex

To identify the vertex, take the values of \(h\) and \(k\) from the vertex form of the function, which is \(-(x-h)^{2}+k\). So, the vertex is \(h=0.5, k=-6.75\), or \((0.5, -6.75)\)

Step 3: Determine the maximum or minimum point

If the leading coefficient \(a < 0\), the function opens downward and the vertex is a maximum point. If the leading coefficient \(a > 0\), the function opens upward and the vertex is a minimum point. In this case, \(a = -1\) (negative), so the vertex is a maximum point.

Key Concepts

Completing the Square

Completing the square is a method used to transform a standard quadratic equation into vertex form. To do this, first rearrange your equation to separate the quadratic and linear terms from the constant. In our example, we start with the equation: \[ g(x) = -x^2 + x - 7 \] We can factor out the negative sign and rearrange the equation: \[ g(x) = -(x^2 - x) - 7 \] Next, make the quadratic expression inside the parentheses a perfect square trinomial by adding and subtracting \( (b/2)^2 \). Here, \( b \) is the coefficient of \( x \), which is \( 1 \). Thus, \( b/2 = 0.5 \), and \( (b/2)^2 = 0.25 \). Add \( 0.25 \) inside the parentheses and compensate it outside: \[ g(x) = -(x^2 - x + 0.25) - 7 + 0.25 \] Now, we have a perfect square trinomial. We can express it in squared notation: \[ g(x) = -(x-0.5)^2 - 6.75 \] This equation now reflects the vertex form of the quadratic function, making it easier to analyze.

Vertex Form

The vertex form of a quadratic function makes it simple to identify important features, such as the vertex of the parabola. The vertex form is given by the equation: \[ f(x) = a(x-h)^2 + k \] Where \( (h, k) \) is the vertex of the parabola. In this context, our transformed function \( g(x) = -(x-0.5)^2 - 6.75 \) shows:

• \( a = -1 \)
• \( h = 0.5 \)
• \( k = -6.75 \)

Thus, the vertex of the parabola is located at \( (0.5, -6.75) \). This form is very useful for sketching the graph of the quadratic as it directly reveals the vertex's position, allowing us to understand the parabola's orientation and shifts.

Parabola Characteristics

Exploring the characteristics of a parabola helps us determine how it behaves. The key characteristics to consider are the vertex, direction of opening, and the maximum or minimum nature of the vertex. For the equation \( g(x) = -(x-0.5)^2 - 6.75 \), the leading coefficient \( a \) is \(-1\). This negatively signed \( a \) indicates that the parabola opens downward. As a result:

• The vertex \((0.5, -6.75)\) is a maximum point because the parabola is inverted.
• Opening downward means any value of \( x \) beyond the vertex will produce smaller \( y \)-values.

Understanding these features helps in graphically visualizing the equation, predicting the function's range and domain, and recognizing real-world applications where a maximum point happens, such as in projectile motion.