Problem 55 - Algebra 2

Question

CHALLENGE Given \(y=a x^{2}+b x+c\) with \(a \neq 0\) , derive the equation for the axis of symmetry by completing the square and rewriting the equation in the form \(y=a(x-h)^{2}+k\)

Step-by-Step Solution

Answer

The axis of symmetry is \(x = -\frac{b}{2a}\).

Step 1: Identify the Quadratic Formula

We start with the given quadratic equation: \[ y = ax^2 + bx + c \] where \(a eq 0\). Our goal is to rewrite this equation in the vertex form \(y = a(x-h)^2 + k\), where \(h\) will give us the axis of symmetry \(x = h\).

Step 2: Factor Out the Leading Coefficient

Factor \(a\) out of the first two terms:\[ y = a(x^2 + \frac{b}{a}x) + c \] This step prepares the expression inside the parentheses for completing the square.

Step 3: Complete the Square

To complete the square inside the parentheses, take half the coefficient of \(x\), square it, and add and subtract it within the parentheses:\[ y = a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right) + c \]Simplifying inside the parentheses:\[ y = a\left((x + \frac{b}{2a})^2 - \left(\frac{b}{2a}\right)^2\right) + c \]

Step 4: Simplify the Equation

Distribute the \(a\) and simplify the equation:\[ y = a(x + \frac{b}{2a})^2 - a\left(\frac{b}{2a}\right)^2 + c \]Simplify further:\[ y = a(x + \frac{b}{2a})^2 + \left(c - \frac{b^2}{4a}\right) \]

Step 5: Identify the Vertex Form and Axis of Symmetry

From the simplified expression:\[ y = a(x + \frac{b}{2a})^2 + \left(c - \frac{b^2}{4a}\right) \]We see the vertex form of a quadratic equation \(y = a(x-h)^2 + k\) where \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\).Thus, the axis of symmetry is given by the equation \(x = h = -\frac{b}{2a}\).

Key Concepts

Quadratic Equation

A quadratic equation is a beautiful expression of mathematics, represented generally as \( y = ax^2 + bx + c \). This equation forms a parabolic curve on a graph, and it’s characterized by the variable raised to the second power, making it a quadratic. The constants \( a \), \( b \), and \( c \) are real numbers, with \( a eq 0 \) to ensure it’s a true quadratic.

• **Coefficient \( a \)**: Controls the "width" and the "direction" of the parabola (upward for positive, downward for negative).
• **Coefficient \( b \)**: Influences the positioning of the vertex horizontally.
• **Constant \( c \)**: Represents the y-intercept of the parabola.

Understanding how each component influences the graph is crucial when you are working on completing the square and rewriting quadratic equations.

Vertex Form

The vertex form of a quadratic equation offers a clear perspective into the shape and position of a parabola on a graph. It’s expressed as \( y = a(x-h)^2 + k \), where the vertex of the parabola is located at the point \((h, k)\).

• **Ease of Graphing**: By converting a quadratic to vertex form, you significantly simplify the graphing process.
• **Characteristics**: The term \( (x - h)^2 \) shifts the parabola horizontally, while \( k \) shifts it vertically.
• **Vertex**: The point \((h, k)\), making it easy to identify maximum or minimum points in the parabola.

To switch from the standard form \( y = ax^2 + bx + c \) to vertex form, mathematicians often use the method of completing the square.

Axis of Symmetry

The axis of symmetry is a vertical line that bisects the parabola into two mirror-image halves. In the vertex form \( y = a(x-h)^2 + k \), this line is defined by \( x = h \).

• **Equation**: Derived using \( x = -\frac{b}{2a} \) from the quadratic equation.
• **Purpose**: Offers insight into the shape and balance of the parabola.
• **Symmetry**: Important for solving and graphing because it gives a line of reflection for the curve.

Completing the square helps us precisely identify this axis, offering a pathway to further understand the nature of quadratic functions.

Factoring

Factoring is a powerful mathematical tool that breaks down equations into simpler parts, and for quadratics, it can often transform expressions into the form \((px + q)(rx + s)\). However, when not easily factorable, completing the square becomes crucial, especially for rewriting in vertex form.

• **Simplifying**: By factoring, you can sometimes go straight to roots without complex algebra.
• **Completing the Square**: When directly factoring isn't feasible, it comes in handy for transformation purposes.
• **Algebraic Approach**: Blends factoring and completing the square to achieve a clean vertex form.

This method does not just clarify the quadratic's properties but also aids in deriving solutions to equations.

Vertex

The vertex is like the pinnacle or the lowest point of a parabola, and understanding its location is integral to interpreting the graph of a quadratic. In vertex form, it’s clearly denoted as \((h, k)\).

• **Much Ado About \((h,k)\)**: Helps define whether the parabola opens upwards or downwards.
• **Highest or Lowest Point**: This point is either a maximum or minimum value of the quadratic function.
• **Turning Point**: It’s where the graph swells changes direction, making it pivotal for analysis.

By understanding how to find and interpret the vertex, you can gain meaningful insights into the quadratic's behavior and features.