Problem 9 - A Graphical Approach to Precalculus with Limits

Question

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}-6 x$$

Step-by-Step Solution

Answer

The function is \(P(x) = (x-3)^2 - 9\) with vertex (3, -9).

Step 1: Completing the Square

Given the function \(P(x) = x^2 - 6x\), we first complete the square to rewrite it in the form \(P(x) = a(x-h)^2 + k\). We take half of the coefficient of \(x\), which is \(-6\), divide it by 2 to get \(-3\), and then square this to get \(9\). Thus, we have:\(P(x) = x^2 - 6x + 9 - 9\).This can be rewritten as:\(P(x) = (x - 3)^2 - 9\).The function is now in the form \(P(x) = a(x-h)^2 + k\), where \(a = 1\), \(h = 3\), and \(k = -9\).

Step 2: Finding the Vertex

The vertex form of a quadratic is \(P(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex. From Step 1, we found \(P(x) = (x-3)^2 - 9\). Thus, the vertex of the parabola is \((3, -9)\).

Step 3: Graphing the Function

To graph \(P(x) = (x-3)^2 - 9\), start by plotting the vertex at the point \((3, -9)\). Since the coefficient \(a = 1\) is positive, the parabola opens upwards. Plot additional points to the left and right of the vertex to ensure the shape, such as \((2, -8)\) and \((4, -8)\), verifying symmetry. Connect these points to form the parabola.

Key Concepts

Completing the Square

Completing the square is a technique used to transform a standard quadratic equation into a form that is easily analyzed. This is useful in graphing and finding key features like the vertex of the parabola. Take the quadratic function \(P(x) = x^2 - 6x\). Begin by identifying the coefficient of the linear term, here it is \(-6\). Take half of this coefficient, which is \(-3\), and square it to get \(9\).
This value, \(9\), is added and subtracted within the function to allow restructuring. Thus, the function \(P(x) = x^2 - 6x\) becomes \(P(x) = x^2 - 6x + 9 - 9\), simplifying to \(P(x) = (x - 3)^2 - 9\).
The function is now expressed in vertex form \(P(x) = a(x-h)^2 + k\), where \(a = 1\), \(h = 3\), and \(k = -9\). This method effectively isolates the perfect square trinomial, simplifying further analysis.

Vertex Form

The vertex form of a quadratic function is \(P(x) = a(x-h)^2 + k\). This form is particularly advantageous because it quickly reveals the vertex of the parabola, an essential feature for graphing. In our example, after completing the square, we have \(P(x) = (x - 3)^2 - 9\).
The vertex \((h, k)\) directly emerges as \((3, -9)\). The vertex tells us the highest or lowest point on the graph depending on the direction of the parabola. Here, since \(a = 1\) (positive), it opens upwards.

• \((h, k)\) indicates the turning point of the parabola.
• Knowing \(a\) helps determine the direction of the parabola's opening.

Recognizing the vertex form enables quick identification and graphing without further complex calculations.

Graphing Quadratics

Graphing a quadratic function once in vertex form is straightforward and visual. Begin by locating the vertex of the parabola. For \(P(x) = (x-3)^2 - 9\), the vertex is at \((3, -9)\). This becomes your first point.
Next, note the value of \(a\) (in our case, \(a = 1\)), which dictates the direction of the parabola's opening. A positive \(a\) implies an upward opening. After plotting the vertex, choose symmetrical points around this vertex to maintain the parabola's shape. For instance, when \(x = 2\), \(P(x) = -8\), and similarly for \(x = 4\).

• Vertex: Plot the vertex point first.
• Symmetrical Points: Select points equidistant from the vertex for accurate shape.
• Connected Curve: Draw a smooth curve through the points to complete the parabola.

This method ensures your graph reflects the quadratic's properties accurately.