Problem 10 - 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. $$P(x)=x^{2}+4 x$$

Step-by-Step Solution

Answer

The function in vertex form is \( P(x) = (x+2)^2 - 4 \) with vertex \((-2, -4)\). The graph is a parabola opening upwards.

Step 1: Complete the Square

To convert the quadratic function into vertex form, we start with the expression for the quadratic function: \[ P(x) = x^2 + 4x \]First, identify the coefficient of the linear term, which is 4. Take half of this coefficient, which is 2, and then square it, resulting in 4. Add and subtract this square within the function to create a perfect square:\[ P(x) = x^2 + 4x + 4 - 4 \]This can be rewritten as:\[ P(x) = (x + 2)^2 - 4 \]

Step 2: Identify the Vertex

Now that the function is in the form \[ P(x) = a(x-h)^2 + k \]we can see that \( a = 1 \), \( h = -2 \), and \( k = -4 \). The vertex form of the function directly provides the vertex of the parabola:The vertex is \( (-2, -4) \)

Step 3: Graph the Function

To graph the function, plot the vertex \( (-2, -4) \) on the coordinate plane. Since the coefficient \( a = 1 \) is positive, the parabola opens upwards. The axis of symmetry is \( x = -2 \).From the vertex, select a few x-values to determine corresponding y-values. For example, - When \( x = -1 \), \[ P(-1) = (-1 + 2)^2 - 4 = 1 - 4 = -3 \]- When \( x = 0 \), \[ P(0) = (0 + 2)^2 - 4 = 4 - 4 = 0 \]Use these points, along with the vertex, to sketch the graph of the parabola. Draw a symmetric curve passing through these points with the vertex at the lowest point because it opens upwards.

Key Concepts

Vertex Form

Quadratic functions are often expressed in the vertex form, which is beneficial for graphing and understanding the properties of the function. The vertex form of a quadratic function is \[P(x) = a(x-h)^2 + k\] where \(h\) and \(k\) are the coordinates of the parabola's vertex, and \(a\) determines the direction and width of the parabola. Rewriting a quadratic in vertex form involves identifying these parameters, which simplifies finding the vertex and other properties of the parabola directly.

• The term \((x-h)^2\) indicates a horizontal shift to the right by \(h\) units. If \(h\) is negative, the shift is to the left.
• The \(k\) in the expression results in a vertical shift by \(k\) units.
• The coefficient \(a\) affects how "open" or "closed" the parabola is. If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

To convert from standard to vertex form, the method of completing the square is often used. This leads directly into the next core concept: Completing the Square.

Completing the Square

Completing the square is a technique used to convert a quadratic equation from its standard form to vertex form. This method involves creating a perfect square trinomial and is essential for understanding and graphing quadratic functions.Here's a step-by-step on completing the square:1. Start with the quadratic equation in standard form, for example, \(P(x) = x^2 + 4x\).2. Take the coefficient of the \(x\) term (here, it's 4), divide it by 2, and square the result: \( (4/2)^2 = 4\).3. Add and subtract this square inside the function: \[P(x) = x^2 + 4x + 4 - 4\]4. Then, factor the perfect square trinomial: \[P(x) = (x + 2)^2 - 4\]This restructuring shows us that \((x+2)^2\) is a perfect square, allowing the function to be rewritten in the form \((x-h)^2 + k\). The function is now readily in vertex form, making the process of finding the vertex and graphing the parabola more straightforward.

Graphing Parabolas

Graphing a quadratic function, specifically a parabola, involves identifying key features like the vertex, the direction of opening, and the axis of symmetry. Once a quadratic is in the vertex form \( a(x-h)^2 + k \), graphing becomes quite systematic.

• First, plot the vertex \( (h, k) \). In our example, the vertex is \((-2, -4)\).
• The coefficient \(a\) dictates the direction: if \(a > 0\), draw the parabola opening upwards. If \(a < 0\), it opens downwards.
• The axis of symmetry is a vertical line passing through the vertex, \(x = h\). For our function, the axis of symmetry is \(x = -2\).
• Plot additional points by choosing x-values around the vertex. Calculate corresponding y-values using the quadratic equation. For example, if \(x = -1\), \(P(-1) = -3\), and if \(x = 0\), \(P(0) = 0\). These points help in accurately drawing the parabola.

Once these points are plotted, sketch a smooth symmetrical curve through them, ensuring the curve is U-shaped if it opens upwards and ∩-shaped if it opens downwards. This visually represents the quadratic function and aids in understanding its behavior and properties on a graph.