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

Step-by-Step Solution

Answer

(a) \( P(x) = -4(x - \frac{1}{2})^2 + 1 \); (b) Vertex: \( (\frac{1}{2}, 1) \); (c) Graph opens downwards.

Step 1: Identify the Standard Form

The given quadratic function is \( P(x) = -4x^2 + 4x \). We want to rewrite it in the vertex form \( P(x) = a(x-h)^2 + k \). First, identify \( a = -4 \), which is the coefficient of \( x^2 \).

Step 2: Complete the Square

To complete the square, we take the coefficient of \( x \), which is \( 4 \), divide it by \( 2 \), and square it. \((\frac{4}{2})^2 = 4 \). Add and subtract this inside the equation to balance it: \(-4(x^2 - x) = -4((x - \frac{1}{2})^2 - (\frac{1}{2})^2) = -4(x - \frac{1}{2})^2 + 1 \).

Step 3: Write the Vertex Form

The function can now be expressed as \( P(x) = -4(x - \frac{1}{2})^2 + 1 \). This is the required format of \( a(x-h)^2 + k \) where \( a = -4 \), \( h = \frac{1}{2} \), and \( k = 1 \).

Step 4: Identify the Vertex

The vertex \((h, k)\) can be directly read from the vertex form \( P(x) = -4(x - \frac{1}{2})^2 + 1 \). Therefore, the vertex of the parabola is \( (\frac{1}{2}, 1) \).

Step 5: Graph the Function

Starting from the vertex \((\frac{1}{2}, 1)\), plot the vertex on the coordinate plane. Since \( a = -4 \), the parabola opens downward, and it is vertically stretched by a factor of 4. Mark a few points on either side of the vertex to complete the parabola's shape, reflecting the steepness.

Key Concepts

Vertex Form

In the realm of quadratic functions, the vertex form is a very convenient expression. It is a different representation of a quadratic equation that makes it easier to identify certain features of the graph, particularly the vertex. The vertex form of a quadratic equation is given by:

• \( P(x) = a(x-h)^2 + k \)

Here,

• \( a \) determines the direction and width of the parabola.
• \( (h, k) \) is the vertex of the parabola.

This format allows us to quickly see the vertex because it is presented directly in the equation. Additionally, if \( a \) is greater than 0, the parabola opens upwards. Conversely, if \( a \) is less than 0, as in our exercise, the parabola opens downwards.

Completing the Square

Completing the square is a mathematical technique used to convert a quadratic equation from its standard form to vertex form. This process involves making a perfect square trinomial from the quadratic formula. Consider the steps outlined in the solution:

• First, take the coefficient of \( x \), halve it, and square the result: \( \left(\frac{4}{2}\right)^2 = 4 \).
• Add and subtract this squared value inside the expression for the quadratic so it becomes a perfect square trinomial.
• This is how the expression \( -4(x^2-x) \) transforms into \( -4((x - \frac{1}{2})^2 - \frac{1}{4}) \).

By completing the square, the function reformulates to the vertex form, making it easier to find the vertex and graph the function.

Graphing Parabolas

Graphing parabolas derived from quadratic functions involves several key steps that help visualize the curve effectively. Start with identifying the vertex from the equation in vertex form. For example, given \( P(x) = -4(x - \frac{1}{2})^2 + 1 \), the vertex is \( (\frac{1}{2}, 1) \).

• Plot the vertex on the graph.
• Determine the direction of the parabola. If \( a < 0 \), it opens downward.
• Consider the value of \( a \) to understand the parabola's steepness. Here, \( a = -4 \) indicates a downward stretch.

After the vertex is plotted, add other points by choosing various values of \( x \), using symmetry relative to the vertex to ensure it looks accurate. This requires understanding how the coefficient \( a \) affects the width and direction of the graph.

Vertex of Parabola

The vertex of a parabola is a crucial aspect of studying quadratic functions. It is the point at which the curve changes direction. Given in the vertex form of an equation \( P(x) = a(x-h)^2 + k \), the vertex is represented by the coordinate \( (h, k) \).

• The vertex helps understand the maximum or minimum value of the quadratic function.
• For parabolas facing upwards, the vertex is the lowest point (minimum).
• For parabolas facing downwards, as in our example, the vertex is the highest point (maximum).

In the specific case of \( P(x) = -4(x - \frac{1}{2})^2 + 1 \), the vertex \( (\frac{1}{2}, 1) \) indicates the peak of the parabola. This understanding is pivotal when analyzing parabolic functions, as it pinpoints the extremum within the context of the function.