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

Step-by-Step Solution

Answer

Vertex form: \( P(x) = (x+2)^2 - 4 \); Vertex: \((-2, -4)\).

Step 1: Write the Quadratic in Vertex Form

The goal is to rewrite the function in the form \( P(x) = a(x-h)^2 + k \). We start with the given equation \( P(x) = x^2 + 4x \). To convert this into the vertex form, we complete the square. First, factor out the coefficient of \( x^2 \), which is 1 in this case, so it remains \( x^2 + 4x \).Next, to complete the square, take half the coefficient of \( x \), which is 4, divide it by 2 to get 2, then square it to get 4. Add and subtract this square inside the expression: \[ P(x) = (x^2 + 4x + 4) - 4 \].This becomes \( P(x) = (x+2)^2 - 4 \) after rewriting \( x^2 + 4x + 4 \) as \( (x+2)^2 \). Thus, the function in vertex form is \( P(x) = 1(x+2)^2 - 4 \).

Step 2: Identify the Vertex of the Parabola

The function is now in vertex form, \( P(x) = a(x-h)^2 + k \), where \( a = 1 \), \( h = -2 \), and \( k = -4 \). The vertex of the parabola is given by the point \( (h, k) \).Therefore, the vertex of the parabola is \( (-2, -4) \).

Step 3: Graph the Function (Conceptual Guidance)

Though we aren't drawing the graph, the general approach is:1. Plot the vertex \((-2, -4)\).2. Since the coefficient \(a=1\) (positive), the parabola opens upwards.3. The axis of symmetry is the vertical line that passes through the vertex, given by \(x = -2\).4. To get more points, choose values of \(x\) around the vertex, such as \(x = -3, -1, 0\), and compute \( P(x) \) for these \(x\)-values using \(P(x) = x^2 + 4x\). Plot these points to form the parabola visually.

Key Concepts

Vertex Form

The vertex form of a quadratic function is a special way of writing the equation which makes it easier to understand the graph's characteristics. It's written as \( P(x) = a(x-h)^2 + k \). Here, \( (h, k) \) represents the vertex of the parabola, and \( a \) affects the width and direction of the parabola's opening. A positive \( a \) means the parabola opens upwards, while a negative \( a \) indicates it opens downwards. This form makes it straightforward to quickly identify the vertex and understand the basic shape of the graph.

Completing the Square

Completing the square is a technique used to transform a standard quadratic equation into vertex form. Involves a few simple algebraic steps:

• Start with your quadratic expression, for example, \( x^2 + 4x \).
• Take the coefficient of the \( x \)-term and divide it by 2, then square it. Here, half of 4 is 2, and squaring 2 gives 4.
• Add and subtract this square within the expression to maintain the equation: \( (x^2 + 4x + 4) - 4 \).
• Factor the perfect square trinomial, resulting in \( (x+2)^2 \), and adjust the equation: \( (x+2)^2 - 4 \).

This process turns \( P(x) = x^2 + 4x \) into \( P(x) = (x+2)^2 - 4 \), converting the equation into vertex form and directly revealing the vertex \( (-2, -4) \).

Graphing Parabolas

Graphing a parabola involves understanding its shape and features. Once in vertex form, the graphing process becomes straightforward:

• Start with plotting the vertex, clearly identified from the vertex form.
• Identify whether the parabola opens upward or downward. This is dictated by the sign of \( a \), which is positive in our example, so the parabola opens upwards.
• Draw the axis of symmetry, a vertical line passing through the vertex.
• Select a few x-values around the vertex, calculate their corresponding y-values, and plot these points.

By connecting the points in a smooth curve, you'll trace the accurate path of the parabola.

Parabola Vertex

The vertex of the parabola is a key point that represents the minimum or maximum of the graph, depending on the direction in which the parabola opens. In the vertex form equation \( P(x) = a(x-h)^2 + k \), the vertex is the point \((h, k)\). It's crucial because:

• The vertex tells you the location of the parabola's peak or valley.
• It is the point where the graph changes direction.
• It lies on the axis of symmetry, making it the most central point of the graph.

For our function, \( P(x) = (x+2)^2 - 4 \), the vertex is \((-2, -4)\), indicating that this point is the lowest point since our parabola opens upwards.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex, effectively splitting the parabola into two mirror-image halves. In the vertex form equation, \( P(x) = a(x-h)^2 + k \), the axis of symmetry is represented by the equation \( x = h \). It offers these insights:

• Helps in visualizing the parabola's structure and symmetry.
• It guides the correct placement of the parabola on a graph.
• Through the line \( x = -2 \) in our problem, we see this line acts as the middle point for both sides of the parabola to mirror each other perfectly.

Understanding the axis of symmetry is crucial for accurately sketching and analyzing the parabola's graph.