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Magazine Chapter 3: Problem 10
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. $$P(x)=x^{2}+4 x$$
Short Answer
Expert verified
Vertex form: \( P(x) = (x+2)^2 - 4 \); Vertex: \((-2, -4)\).
Step by step solution
01
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 \( P(x) = 1(x+2)^2 - 4 \).
02
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 \( (-2, -4) \).
03
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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: \( (x+2)^2 - 4 \).
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.
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.
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.
