Problem 9 - A Graphical Approach to College Algebra

Question

For each quadratic function defined , (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 \((x - 3)^2 - 9\), with vertex (3, -9), opening upwards.

Step 1: Complete the Square

To rewrite the quadratic function in the vertex form \( P(x) = a(x-h)^2 + k \), we need to complete the square.Start with the given function: \( P(x) = x^2 - 6x \).The process of completing the square involves creating a perfect square trinomial from the quadratic expression \( x^2 - 6x \).1. Identify the coefficient of \( x \), which is -6.2. Divide this by 2 to get -3, then square it to get 9.3. Add and subtract 9 inside the equation: \( P(x) = (x^2 - 6x + 9) - 9 \).4. This can be rewritten as \( P(x) = (x - 3)^2 - 9 \).So, the function in vertex form is \( P(x) = (x - 3)^2 - 9 \).

Step 2: Identify the Vertex

The vertex form of a quadratic function is \( P(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex.In our function \( P(x) = (x-3)^2 - 9 \), we see that \( h = 3 \) and \( k = -9 \).Therefore, the vertex of the parabola is \( (3, -9) \).

Step 3: Graph the Function

To graph the function \( P(x) = (x-3)^2 - 9 \), follow these steps:1. Plot the vertex \((3, -9)\) as the starting point on the graph.2. Since the coefficient of \( (x-h)^2 \) is positive, the parabola opens upwards.3. Determine additional points by choosing x-values around the vertex, such as \( x = 2 \) and \( x = 4 \), which will be symmetric around the vertex.4. Calculate \( P(2) = (2-3)^2 - 9 = -8 \) and \( P(4) = (4-3)^2 - 9 = -8 \).5. Plot the points \((2, -8)\) and \((4, -8)\), and draw a smooth curve through these points, opening upwards from the vertex.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, which helps in rewriting it in vertex form. Consider the quadratic function given: \[ P(x) = x^2 - 6x \] To complete the square, focus on the quadratic and linear terms, in this case, \(x^2 - 6x\). Follow these steps:

• Identify the coefficient of \(x\), which is \(-6\).
• Divide it by 2 to get \(-3\), then square this result to obtain \(9\).
• Add and subtract \(9\) inside the function, forming: \[ P(x) = (x^2 - 6x + 9) - 9 \]

Next, recognize that \((x^2 - 6x + 9)\) is a perfect square trinomial and rewrite it as \((x-3)^2\). The quadratic equation is now in vertex form:\[ P(x) = (x-3)^2 - 9 \] This transformation makes it easier to graph and identify the vertex of the parabola.

Vertex Form

The vertex form of a quadratic function is a special way of expressing the quadratic equation that highlights its vertex, which is the highest or lowest point on a parabola. It takes the form: \[ P(x) = a(x-h)^2 + k \] Here,

• \(a\) is the coefficient that determines the direction and width of the parabola.
• \((h, k)\) is the vertex and gives the peak or dip of the curve.

For our completed equation \( P(x) = (x-3)^2 - 9 \):

• \(h = 3\) and \(k = -9\), making the vertex \((3, -9)\).
• The parabola opens upwards because \(a\) is positive (implicit 1 here).

The vertex form simplifies understanding how shifting and scaling transform the graph of the quadratic. Understanding the vertex form empowers students to
quickly determine key characteristics of quadratic functions, such as the vertex and direction of opening.

Graphing Parabolas

Graphing parabolas, especially from the vertex form, allows you to visualize quadratic functions with ease. Once the equation is expressed as \[ P(x) = a(x-h)^2 + k \], use this form to sketch the parabola more effectively. Begin by plotting the vertex point. For \( P(x) = (x-3)^2 - 9 \):

• The vertex is at \((3, -9)\), so plot this point first.
• Since \(a>0\), the parabola opens upwards.

Next, select x-values around the vertex to find additional points. Given the function's symmetry, choose x-values a little to the left and right of the vertex like \(x = 2\) and \(x = 4\). Compute the y-values:

• For \(x = 2\), \(P(2) = (2-3)^2 - 9 = -8\).
• For \(x = 4\), \(P(4) = (4-3)^2 - 9 = -8\).

Plot these points, reflecting across the vertex to maintain symmetry. Lastly, draw a smooth curve through these points, ensuring the parabola is opening upwards from the vertex. By following these steps, the graph not only becomes accurate but visually intuitive, aiding deeper understanding of the curve's characteristics.