Problem 13 - 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. $$P(x)=-2 x^{2}+6 x$$

Step-by-Step Solution

Answer

(a) Vertex form: \(P(x) = -2(x - \frac{3}{2})^2 + \frac{9}{2}\); (b) Vertex: \((\frac{3}{2}, \frac{9}{2})\); (c) Graph is a downward parabola.

Step 1: Standard Form Identification

The given quadratic function is \(P(x) = -2x^2 + 6x\). This equation is in the standard form \(ax^2 + bx + c\) where \(a = -2\), \(b = 6\), and \(c = 0\).

Step 2: Complete the Square

To convert the quadratic function into vertex form \(P(x) = a(x-h)^2 + k\), we complete the square. Start by factoring out \(-2\) from the first two terms: \(-2(x^2 - 3x)\).

Step 3: Find the Perfect Square

To complete the square inside the parentheses, take half of the coefficient of \(x\) (which is \(-3\) divided by \(2\)), square it, and add and subtract that square inside the parentheses: \(-2(x^2 - 3x + \frac{9}{4} - \frac{9}{4})\).

Step 4: Simplify the Expression

The expression inside the parentheses \((x^2 - 3x + \frac{9}{4})\) is a perfect square trinomial. Rewrite it as \((-2((x - \frac{3}{2})^2 - \frac{9}{4}))\).

Step 5: Distribute and Simplify

Distribute the \(-2\) into the expression: \(-2(x - \frac{3}{2})^2 + \frac{9}{2}\). So, the function in vertex form is \(P(x) = -2(x - \frac{3}{2})^2 + \frac{9}{2}\).

Step 6: Identify the Vertex

In the vertex form \(P(x) = a(x-h)^2 + k\), \(h = \frac{3}{2}\) and \(k = \frac{9}{2}\). Therefore, the vertex of the parabola is \((\frac{3}{2}, \frac{9}{2})\).

Step 7: Graph the Function

To graph the function, plot the vertex \((\frac{3}{2}, \frac{9}{2})\) on a graph. Since \(a = -2\) is negative, the parabola opens downward. Plot additional points, such as the y-intercept \((0, 0)\), and mirror them across the axis of symmetry \(x = \frac{3}{2}\) for accuracy.

Key Concepts

Vertex Form

The vertex form of a quadratic function is a popular way to express parabolas, particularly because it highlights the vertex, making graphing easier. In the expression \(P(x)=a(x-h)^2+k\),

• \(a\) represents the stretch or compression factor, as well as whether the parabola opens upward (\(a>0\)) or downward (\(a<0\)).
• \((h, k)\) is the vertex of the parabola.

This form is highly useful for identifying key features of the parabola, such as its vertex and direction. By comparing it to the standard form \(ax^2 + bx + c\), we can transform it into vertex form by completing the square, a method particularly effective for simplifying the graphing process.

Completing the Square

Completing the square is a fundamental technique for converting quadratic functions from standard form to vertex form. This process involves:

• First, isolating the quadratic and linear terms.
• Then, determining the value needed to complete the square, which is done by taking half the linear coefficient, squaring it, adding and subtracting it inside the function.

For example, given \(P(x) = -2x^2 + 6x\), you first factor out \(-2\) from the quadratic and linear term: \(-2(x^2 - 3x)\). Then, you compute the square needed: \((\frac{-3}{2})^2 = \frac{9}{4}\), and adjust the equation accordingly. Completing the square simplifies to a perfect square trinomial, allowing for an easy rewrite into vertex form.
This technique empowers you to easily identify the vertex and other properties of the parabola.

Parabola Vertex

The vertex of a parabola is the highest or lowest point, depending on its direction. It is located at \((h, k)\) in the vertex form \(P(x) = a(x-h)^2 + k\).
In the exercise, completing the square helped find the function \(P(x) = -2(x - \frac{3}{2})^2 + \frac{9}{2}\), revealing the vertex as \((\frac{3}{2}, \frac{9}{2})\). This point becomes the focus around which the parabola is symmetric. Understanding the vertex is crucial:

• If the parabola opens upwards, \(k\) is the minimum value.
• If it opens downwards, \(k\) is the maximum value.
Knowing the vertex offers insights into the parabola's behavior and aids in accurately plotting its graph.

Graphing Parabolas

Graphing parabolas, especially when they are in vertex form, becomes a straightforward task. Start by plotting the vertex on a coordinate plane; in our case, this is \((\frac{3}{2}, \frac{9}{2})\).
Once the vertex is marked:

• Understand the direction: A negative \(a\) value (as in \(-2\) here) indicates the parabola opens downward.
• Identify the axis of symmetry, which is a vertical line through the vertex \(x = h\).
• Find additional points such as the y-intercept, and mirror them across the axis of symmetry for a balanced plot.

In the given example, the y-intercept is at \((0, 0)\). By plotting these points and sketching the U-shape curve, you can visualize the entire parabola. With practice, graphing becomes an intuitive process that strengthens the understanding of quadratic functions.