Finding Vertex and Axis of Symmetry

Introduction

Key Concepts

1. Understanding Quadratic Functions

A quadratic function is a second-degree polynomial of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a \neq 0$. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient $a$.

2. Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. It represents the maximum or minimum value of the quadratic function.

To find the vertex $(h, k)$ of a quadratic function $f(x) = ax^2 + bx + c$, use the formula: $$ h = -\frac{b}{2a} $$ Once $h$ is found, substitute it back into the function to find $k$: $$ k = f(h) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c $$ Simplifying: $$ k = \frac{4ac - b^2}{4a} $$ Thus, the vertex is at: $$ \left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right) $$

3. Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror images. It passes through the vertex of the parabola.

The equation of the axis of symmetry for the quadratic function $f(x) = ax^2 + bx + c$ is: $$ x = -\frac{b}{2a} $$ This line represents the vertical line that symmetrically splits the parabola.

4. Graphing the Parabola

To graph a quadratic function, follow these steps:

1. Identify the coefficients: Determine the values of $a$, $b$, and $c$ from the quadratic equation.
2. Calculate the vertex: Use the vertex formula to find $(h, k)$.
3. Determine the axis of symmetry: Use $x = -\frac{b}{2a}$.
4. Find additional points: Choose $x$-values on either side of the axis of symmetry and compute the corresponding $f(x)$ to plot more points.
5. Plot the points and draw the parabola: Plot the vertex, axis of symmetry, and additional points, then draw a symmetric parabola through these points.

5. Example: Finding the Vertex and Axis of Symmetry

Let's consider the quadratic function: $$ f(x) = 2x^2 - 4x + 1 $$

• Identify the coefficients: $a = 2$, $b = -4$, $c = 1$
• Calculate the vertex: $$ h = -\frac{b}{2a} = -\frac{-4}{2 \times 2} = \frac{4}{4} = 1 $$ $$ k = f(1) = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1 $$ So, the vertex is at $(1, -1)$.
• Determine the axis of symmetry: $$ x = -\frac{b}{2a} = 1 $$ So, the axis of symmetry is the line $x = 1$.

Using this information, we can graph the parabola with the vertex at $(1, -1)$ and symmetry along $x = 1$.

6. Alternative Forms of Quadratic Functions

Quadratic functions can also be expressed in different forms, such as the vertex form and the standard form. Understanding these forms aids in easily identifying the vertex and the axis of symmetry.

Vertex Form

The vertex form of a quadratic function is: $$ f(x) = a(x - h)^2 + k $$ where $(h, k)$ is the vertex of the parabola. This form makes it straightforward to identify the vertex directly from the equation.

Standard Form

The standard form of a quadratic function is: $$ f(x) = ax^2 + bx + c $$ To convert from standard form to vertex form, complete the square:

• Start with $f(x) = ax^2 + bx + c$
• Factor out $a$ from the first two terms: $$ f(x) = a\left(x^2 + \frac{b}{a}x\right) + c $$
• Complete the square inside the parentheses: $$ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 $$ $$ = \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} $$
• Rewrite the function: $$ f(x) = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c $$ $$ = a\left(x - \left(-\frac{b}{2a}\right)\right)^2 + \left(c - \frac{b^2}{4a}\right) $$

Thus, the vertex form is: $$ f(x) = a\left(x - h\right)^2 + k $$ where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.

7. Applications of Vertex and Axis of Symmetry

Understanding the vertex and axis of symmetry has practical applications in various fields:

• Physics: Modeling projectile motion where the vertex represents the maximum height.
• Engineering: Designing parabolic structures like bridges and satellite dishes.
• Economics: Determining maximum profit or minimum cost in quadratic profit functions.

8. Solving Real-World Problems

Consider a scenario where a company wants to maximize its profit based on the number of units sold. The profit function can often be modeled as a quadratic equation. By finding the vertex, the company can determine the optimal number of units to produce and sell to achieve maximum profit.

9. Graphing Using Technology

Modern graphing calculators and software tools can assist in visualizing quadratic functions. By inputting the function, students can observe the vertex and axis of symmetry graphically, enhancing their understanding of these concepts.

10. Practice Problems

Engaging with practice problems reinforces the concepts of vertex and axis of symmetry. Here are a few examples:

• Problem 1: Find the vertex and axis of symmetry for the quadratic function $f(x) = -3x^2 + 6x - 2$.
• Solution:
  • Identify coefficients: $a = -3$, $b = 6$, $c = -2$
  • Find $h = -\frac{b}{2a} = -\frac{6}{2(-3)} = 1$
  • Find $k = f(1) = -3(1)^2 + 6(1) - 2 = -3 + 6 - 2 = 1$
  • Vertex: $(1, 1)$
  • Axis of symmetry: $x = 1$
• Problem 2: Determine the vertex and axis of symmetry for $f(x) = x^2 - 4x + 4$.
• Solution:
  • Identify coefficients: $a = 1$, $b = -4$, $c = 4$
  • Find $h = -\frac{-4}{2(1)} = 2$
  • Find $k = f(2) = (2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$
  • Vertex: $(2, 0)$
  • Axis of symmetry: $x = 2$

Comparison Table

Summary and Key Takeaways