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Chapter 3: Problem 63

Derive the formula \(x=\frac{-b}{2 a}\) for the \(x\) -coordinate of the vertex of parabola \(y=a x^{2}+b x+c\)

Short Answer

Expert verified
The derived formula for the vertex's x-coordinate is \(x = \frac{-b}{2a}\). This is obtained from completing the square.

Step by step solution

01

Identify the General Form of a Parabola

The general form of the quadratic equation, which represents a parabola, is \( y = ax^2 + bx + c \). Our task is to find the \( x \)-coordinate of the vertex of this parabola.
02

Use the Formula for the Vertex

A quadratic equation \( y = ax^2 + bx + c \) can be rewritten to identify its vertex. The standard formula for the \( x \)-coordinate of the vertex is given by \( x = \frac{-b}{2a} \). We aim to derive this formula.
03

Completing the Square

Rewrite the quadratic in the form of a completed square to find the vertex. Start with the equation \( y = ax^2 + bx + c \). First, factor \( a \) out of the quadratic terms: \( y = a(x^2 + \frac{b}{a}x) + c \).
04

Identify Completing the Square Term

To complete the square inside the parenthesis, take half of the coefficient of \( x \), which is \( \frac{b}{a} \), and square it. Add and subtract \( \left( \frac{b}{2a} \right)^2 \) within the parentheses: \( y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \).
05

Simplify the Completed Square

Simplify the expression to \( y = a\left( (x + \frac{b}{2a})^2 - \left(\frac{b}{2a}\right)^2 \right) + c \). This expression is now in the form \( y = a(x-h)^2 + k \), where \( h = -\frac{b}{2a} \) and \( k = \text{some number} \).
06

Derive the Vertex's x-coordinate

In the vertex form \( y = a(x-h)^2 + k \), the \( x \)-coordinate of the vertex is \( h \). So, comparing \( y = a(x + \frac{b}{2a})^2 + \text{constant} \), we conclude that \( x = -\frac{b}{2a} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making it easier to analyze. It is especially helpful for identifying key features of a parabola, like its vertex. You begin with a standard quadratic equation, such as \(y = ax^2 + bx + c\). The goal is to express this complex quadratic as a simpler expression that naturally reveals the vertex.
  • Start by factoring out the coefficient \(a\) from the quadratic and linear terms: \(y = a(x^2 + \frac{b}{a}x) + c\).
  • Next, focus on completing the square inside the parentheses. This means adjusting the expression \(x^2 + \frac{b}{a}x\) so it forms a perfect square trinomial.
  • To do this, calculate \(\left(\frac{b}{2a}\right)^2\) and add and subtract it inside the parentheses. This does not alter the equation fundamentally as the same value is added and subtracted.
  • Mathematically, it looks like: \(y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c\).
  • Then, you simplify it to: \(y = a\left((x + \frac{b}{2a})^2 - \left(\frac{b}{2a}\right)^2\right) + c\).
This transformed equation makes it clear how the parabola shifts on a graph.
Quadratic Equation
Understanding the quadratic equation is essential when exploring parabolas. A general quadratic equation is in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. This equation describes a U-shaped curve called a parabola. Here are some important points:
  • It can open upwards (if \(a > 0\)) or downwards (if \(a < 0\)).
  • The vertex is the point where the parabola reaches its peak or lowest point (depending on the direction it opens).
  • The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
The importance of transforming the quadratic into vertex form (through completing the square) is that it makes the quadratic easier to graph and study. By identifying the vertex, one can easily determine the direction and width of the parabola.
Deriving Formulas
Deriving formulas is a process where mathematical relationships are established through algebraic manipulation. Specifically, when deriving the formula for the \(x\)-coordinate of the vertex of a parabola, you are tasked with converting a standard quadratic equation into a vertex form.
  • Start with the general quadratic equation: \(y = ax^2 + bx + c\).
  • Use the method of completing the square to transform it into vertex form: \(y = a(x-h)^2 + k\), where \(h\) and \(k\) are constants that give information about the vertex.
  • Comparing the complete square form \(y = a(x + \frac{b}{2a})^2 + \text{some number}\) to vertex form helps identify \(h = -\frac{b}{2a}\).
By deducing this from the process above, you derive the exact position of the vertex on the \(x\)-axis without guessing. This derived formula, \(x = -\frac{b}{2a}\), is a straightforward method for quickly finding the vertex of any parabola given its quadratic equation.

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