2-6-5 Quadratic Functions Explained
Key Concepts of Quadratic Functions
Quadratic functions are polynomial functions of the second degree, typically expressed in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \). Key concepts include:
- Standard Form: \( f(x) = ax^2 + bx + c \).
- Vertex Form: \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
- Vertex and Axis of Symmetry: The highest or lowest point on the graph and the vertical line passing through it.
- Graphing Quadratic Functions: Representing the function as a parabola on the coordinate plane.
- Solving Quadratic Equations: Finding the roots or zeros of the function.
1. Standard Form
The standard form of a quadratic function is \( f(x) = ax^2 + bx + c \). The coefficient \( a \) determines the direction and width of the parabola, \( b \) affects the position of the vertex, and \( c \) is the y-intercept.
Example:
Given \( f(x) = 2x^2 + 4x + 1 \):
Here, \( a = 2 \), \( b = 4 \), and \( c = 1 \). The parabola opens upwards because \( a > 0 \).
2. Vertex Form
The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \). This form directly gives the vertex \((h, k)\) and the axis of symmetry \( x = h \).
Example:
Given \( f(x) = -3(x - 1)^2 + 2 \):
The vertex is \((1, 2)\) and the axis of symmetry is \( x = 1 \). The parabola opens downwards because \( a < 0 \).
3. Vertex and Axis of Symmetry
The vertex is the highest or lowest point on the parabola. The axis of symmetry is the vertical line passing through the vertex. For the standard form \( f(x) = ax^2 + bx + c \), the vertex \((h, k)\) can be found using \( h = -\frac{b}{2a} \) and \( k = f(h) \).
Example:
For \( f(x) = 2x^2 + 4x + 1 \):
\[ h = -\frac{4}{2 \cdot 2} = -1 \]
\[ k = f(-1) = 2(-1)^2 + 4(-1) + 1 = 2 - 4 + 1 = -1 \]
The vertex is \((-1, -1)\) and the axis of symmetry is \( x = -1 \).
4. Graphing Quadratic Functions
Graphing a quadratic function involves plotting points and drawing a smooth curve (parabola). The vertex and axis of symmetry are crucial for accurate graphing.
Example:
Graph \( f(x) = x^2 - 4x + 3 \):
Find the vertex: \( h = -\frac{-4}{2 \cdot 1} = 2 \), \( k = f(2) = 2^2 - 4(2) + 3 = 4 - 8 + 3 = -1 \).
Plot the vertex \((2, -1)\) and additional points such as \((0, 3)\), \((1, 0)\), \((3, 0)\), and \((4, 3)\). Connect these points to form a parabola.
5. Solving Quadratic Equations
Solving quadratic equations involves finding the roots or zeros of the function, where \( f(x) = 0 \). This can be done using factoring, completing the square, or the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
Example:
Solve \( x^2 - 4x + 3 = 0 \):
Factor: \( (x - 3)(x - 1) = 0 \).
Set each factor to zero: \( x - 3 = 0 \) or \( x - 1 = 0 \).
The solutions are \( x = 3 \) and \( x = 1 \).
Examples and Analogies
To better understand quadratic functions, consider the following analogy:
Imagine a quadratic function as a roller coaster track. The vertex is the highest or lowest point, and the axis of symmetry is the central support beam. The parabola represents the track, and solving the quadratic equation is like finding the points where the roller coaster touches the ground.
Practical Applications
Quadratic functions are used in various real-world applications, such as modeling projectile motion, analyzing profit and loss in business, and determining the path of a thrown object. Understanding quadratic functions allows for accurate predictions and efficient problem-solving in these contexts.