Problem 5 - 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. $$P(x)=x^{2}-2 x-15$$

Step-by-Step Solution

Answer

The vertex form is \(P(x) = (x-1)^2 - 16\), with vertex \((1, -16)\).

Step 1: Complete the Square

To rewrite the function in the vertex form \(P(x) = a(x-h)^2 + k\), we complete the square for \(P(x) = x^2 - 2x - 15\). First, we focus on the terms with \(x\): \(x^2 - 2x\). To complete the square, take half of the coefficient of \(x\), which is \(-2\), giving \(-1\), and then square it, obtaining \((x-1)^2 = x^2 - 2x + 1\). Hence, we rewrite the expression: \(x^2 - 2x = (x-1)^2 - 1\). Substituting this back, we have \(P(x) = (x-1)^2 - 16\).

Step 2: Write in Vertex Form

Now that we have completed the square in Step 1, we have the expression \((x-1)^2 - 16\). The function in vertex form is \(P(x) = 1(x-1)^2 - 16\), where \(a = 1\), \(h = 1\), and \(k = -16\).

Step 3: Identify the Vertex

The vertex form of a quadratic function is \(P(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola. From our equation \(P(x) = 1(x-1)^2 - 16\), the vertex \((h, k)\) is \((1, -16)\).

Step 4: Graph the Function

Graph the function by plotting the vertex \((1, -16)\) and noting that \(a = 1\) indicates the parabola opens upwards. To sketch, plot additional points by evaluating the function for values near \(x = 1\), such as \(x = 0\) and \(x = 2\), where \(P(0) = (0-1)^2 - 16 = -15\) and \(P(2) = (2-1)^2 - 16 = -15\), confirming symmetry around the vertex. Connect these points to form a parabola.

Key Concepts

Vertex Form

The vertex form of a quadratic function is an extremely useful way to represent the equation of a parabola. It is expressed as \( P(x) = a(x-h)^2 + k \), where:

• \(a\): Determines the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards. If it's negative, it opens downwards.
• \(h\): Represents the horizontal shift of the parabola from the origin.
• \(k\): Represents the vertical shift of the parabola from the origin.

By rewriting a quadratic function in vertex form, we can easily identify the vertex \((h, k)\), which is the highest or lowest point of the parabola, depending on whether it opens up or down.
This form also makes graphing and understanding transformations much easier.

Completing the Square

Completing the square is a method used to transform a standard quadratic equation into the vertex form. This process helps to reveal the characteristics of the quadratic function. Here’s how you complete the square:

• Start with the standard form of the quadratic equation, such as \( P(x) = x^2 - 2x - 15 \).
• Focus on the terms involving \(x\), which are \(x^2 - 2x\).
• Take half of the \(x\)-coefficient (\(-2\)), which is \(-1\), and square it to get \(1\).
• Add and subtract this square inside the equation: \(x^2 - 2x = (x-1)^2 - 1\).
• Substitute this expression back into the equation to get \(P(x) = (x-1)^2 - 16\).

Through completing the square, we've rewritten the quadratic in a form that clearly shows the vertex, facilitating graphing and further analysis.

Parabola Graphing

Graphing a parabola involves plotting its vertex and additional points to illustrate its shape. Here are the steps:

• The vertex is your starting point. For \( P(x) = (x-1)^2 - 16 \), the vertex is at \( (1, -16) \).
• The direction the parabola opens is dictated by the \(a\)-value. Since \(a = 1\) here, the parabola opens upwards.
• To capture the shape, calculate \(P(x)\) for nearby \(x\) points, like \(x = 0\) and \(x = 2\), yielding \(P(0) = -15\) and \(P(2) = -15\).
• These calculations show symmetry about the vertex, helping you sketch the basic U-shape of the parabola.

Understanding these key points allows for accurate plotting and interpreting of the graph, aligning it with the properties derived from the vertex form.

Function Transformation

Function transformations are operations that alter the original function's graph in various ways. When dealing with quadratics, transformations dictate direction, positioning, and orientation:

• Vertical Shifts: The \(k\) value moves the parabola up or down. In \(P(x) = (x-1)^2 - 16\), the parabola shifts 16 units down.
• Horizontal Shifts: The \(h\) value shifts left or right. Here, \(h = 1\) moves the whole parabola 1 unit to the right.
• Vertical Stretching/Compressing: The value of \(a\) affects how "wide" or "narrow" the parabola appears. Since \(a = 1\) in this example, there is no vertical stretching/compressing.

By visualizing these transformations, complex functions become much easier to graph and understand, leading to better intuition about their behaviors and properties.