Problem 14 - 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. $$P(x)=-4 x^{2}+4 x$$

Step-by-Step Solution

Answer

The vertex form is \(P(x) = -4(x-\frac{1}{2})^2 + 1\), with a vertex at \(\left( \frac{1}{2}, 1 \right)\).

Step 1: Identify Coefficients

For the quadratic function given as \(P(x) = -4x^2 + 4x\), identify the coefficients: \(a = -4\), \(b = 4\), and \(c = 0\) (since there is no constant term).

Step 2: Complete the Square

Rewrite the function by completing the square. Start by factoring \(-4\) from the first two terms: \(P(x) = -4(x^2 - x)\). To complete the square inside the parenthesis, take half of the linear term coefficient \(b = -1\), square it to get \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\). Add and subtract \(\frac{1}{4}\) inside the parenthesis: \(P(x) = -4(x^2 - x + \frac{1}{4} - \frac{1}{4})\). This simplifies to: \(P(x) = -4((x - \frac{1}{2})^2 - \frac{1}{4})\).

Step 3: Simplify Completed Square

Distribute the \(-4\) across the terms adjusted for the square: \(P(x) = -4(x - \frac{1}{2})^2 + 1\). This is the function in vertex form, where \(a = -4\), \(h = \frac{1}{2}\), and \(k = 1\).

Step 4: Identify the Vertex

The vertex form \(P(x) = a(x-h)^2+k\) gives the vertex of the parabola at \((h, k)\). Thus, the vertex of our function is \(\left( \frac{1}{2}, 1 \right)\).

Step 5: Graph the Function

Sketch the graph using the vertex. The vertex is \(\left( \frac{1}{2}, 1 \right)\). Since \(a = -4\), the parabola opens downwards and is narrower than the standard parabola \(x^2\) due to the larger magnitude of \(a\). Plot additional points by choosing values of \(x\) nearby to show more curvature. For example, use \(x = 0\) and \(x = 1\) to check more points on the curve.

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that reveals key characteristics about its graph, like its vertex. To complete the square for a quadratic in the form of \( ax^2 + bx + c \), we mainly focus on the terms \( ax^2 + bx \). Here's how:

• Factor out any coefficients from the \( x^2 \) and \( x \) terms.
• Next, take half of the coefficient of \( x \), square it, and add or subtract this square inside the equation to balance it.

By adding and subtracting the same number, you essentially "complete" the square, forming a perfect square trinomial inside the parenthesis. This step simplifies the equation and transforms it into a form that's easier to work with.

Vertex Form of a Parabola

The vertex form of a quadratic equation is given by \( P(x) = a(x - h)^2 + k \). This form is incredibly useful because it allows us to easily identify the vertex of the parabola, a crucial part of graphing quadratics.In this form:

• \( a \) determines the opening direction and width of the parabola.
• \( (h, k) \) represents the vertex, or the "tip" of the parabola.

The vertex form makes it simpler to analyze and graph the function directly since you can read off the vertex and know how the parabola behaves without additional calculations.

Graphing Parabolas

Graphing parabolas involves understanding the orientation and shape of the parabola based on its quadratic equation. The graph of a quadratic function is always a parabola.To graph a quadratic function effectively:

• Identify the vertex, which acts as a central point.
• Determine the direction it opens: upward if \( a > 0 \) or downward if \( a < 0 \).
• Calculate and plot a few additional points on either side of the vertex for accuracy.
• Consider the width, influenced by the absolute value of \( a \); a larger \(|a|\) makes the parabola narrower.

Using these steps, you can sketch the graph even without a calculator, as the vertex form provides all needed information.

Vertex of a Parabola

The vertex of a parabola is crucial because it represents the maximum or minimum point, depending on the parabola's orientation. In the vertex form \( P(x) = a(x-h)^2+k \), the vertex is simply \( (h, k) \).For practical purposes:

• The vertex tells us the "peak" where the parabola changes direction.
• If \( a > 0 \), the vertex is the minimum point; if \( a < 0 \), it's the maximum.
• This point helps in understanding the overall shift and graphing the parabola accurately.

The vertex also helps in solving real-world problems modeled by quadratic functions, as it can indicate optimum values or transition points.