Problem 34 - 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. $$h(x)=x^{2}-4 x+6$$

Step-by-Step Solution

Answer

The quadratic function in the form \(f(x)=a(x-h)^{2}+k\) is \(h(x) = (x - 2)^{2} + 2\), where the vertex of the parabola is \((2, 2)\) which is a minimum point.

Step 1: Rewrite in the Vertex Form

To rewrite the function \(h(x) = x^{2} - 4x + 6\) in the form \(f(x)=a(x-h)^{2}+k\), we need to complete the square. To do so, we place the constant term (6) to the right side of the equation. Hence, \(h(x) - 6 = x^{2} - 4x\). Next, we take half the coefficient of x, square it, and add it to both sides of the equation. Half of -4 is -2, and \((-2)^2 = 4\). Therefore, by adding 4 to both sides we obtain, \(h(x) - 6 + 4 = (x^{2} - 4x + 4)\) which simplifies to \(h(x) - 2 = (x - 2)^2\). Bringing back the constants on one side, we rewrite the equation in the vertex form as \(h(x) = (x - 2)^2 + 2\).

Step 2: Find the Vertex

The vertex of a function in the form \(f(x) = a(x - h)^{2} + k\) is \((h , k)\). Thus, for the function \(h(x) = (x - 2)^{2} + 2\), the vertex is \((2, 2)\).

Step 3: Determine the Minimum or Maximum

The vertex will be a minimum or maximum point depending on the sign of the coefficient 'a' in the form \(f(x) = a(x - h)^{2} + k\). If 'a' is greater than 0, then the parabola opens upwards and so the vertex is a minimum point. In the case of the function \(h(x) = (x - 2)^{2} + 2\), 'a' is 1 which is greater than 0. As such, the vertex is a minimum point.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function that is characterized by the highest exponent being two. Written in the standard form, it looks like:

• \( f(x) = ax^2 + bx + c \)

where \( a, b, \) and \( c \) are constants, and \( x \) is the variable. This form displays the typical parabola shape of a quadratic function on a graph.
The quadratic function is central to various topics in mathematics, including algebra and calculus, because of its ability to model real-world phenomena such as projectile motion, area computations, and more.

Vertex Form

The vertex form of a quadratic function provides a clearer depiction of its graph. In this form, the function is expressed as:

• \( f(x) = a(x-h)^2 + k \)

Here, \( (h, k) \) represents the vertex of the parabola. The parameter \( a \) influences the width and direction of the parabola.
Completing the square is a method employed to transition a standard quadratic equation into vertex form. This process involves adjusting the equation to reveal the perfect square trinomial, which directly relates to the vertex of the parabola. Once in vertex form, determining the vertex and understanding the parabola's behavior becomes swift and intuitive.

Parabola Vertex

The vertex of a parabola is a very significant point. It is the spot where the parabola changes direction and represents either the maximum or minimum point of the function. In vertex form, the vertex can be effortlessly identified as \( (h, k) \).

• The x-coordinate \( h \) indicates the parabola’s line of symmetry.
• The y-coordinate \( k \) shows the parabola’s highest or lowest point.

By identifying the vertex, we gain more insights into the function's graph. This allows us to easily sketch the graph of the quadratic function or solve optimization problems related to the function.

Maximum and Minimum Points

Understanding maximum and minimum points is vital when analyzing quadratic functions. Depending on the value of \( a \) in the vertex form \( f(x) = a(x-h)^2 + k \), the orientation of the parabola changes:

• If \( a > 0 \), the parabola opens upwards, and the graph's lowest point, or minimum, is at the vertex \( (h, k) \).
• If \( a < 0 \), the parabola opens downwards, and the vertex becomes the highest point or maximum.

In optimization contexts, these points can represent crucial values, such as the lowest cost, shortest time, or maximum score. Recognizing whether your quadratic function provides a maximum or minimum helps in problem-solving and decision-making scenarios.