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Questions and Answers
What is the highest power of the variable in a quadratic function?
2
What is the standard form of a quadratic function?
f(x) = ax^2 + bx + c
What determines the direction of the opening of the parabola?
The value of a
What is the axis of symmetry?
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What are the two methods for solving quadratic equations?
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What is the formula for the quadratic formula?
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What type of real-world scenarios are modeled using quadratic functions?
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What is the vertex of the parabola?
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Study Notes
Quadratic Functions
Definition
A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two.
Standard Form
The standard form of a quadratic function is:
f(x) = ax^2 + bx + c
where:
- a, b, and c are constants
- a ≠0 (if a = 0, it's not a quadratic function)
- x is the variable
Graph
The graph of a quadratic function is a parabola that opens:
- Upwards if a > 0
- Downwards if a < 0
The vertex of the parabola is the lowest or highest point, depending on the direction of the opening.
Key Features
- Axis of Symmetry: The vertical line that passes through the vertex, dividing the parabola into two identical halves.
- Vertex: The lowest or highest point on the parabola.
- x-intercepts: The points where the parabola intersects the x-axis.
- y-intercept: The point where the parabola intersects the y-axis.
Solving Quadratic Equations
Quadratic equations can be solved using:
- Factoring: If the equation can be written in the form (x - r)(x - s) = 0, then the solutions are x = r and x = s.
- Quadratic Formula: If factoring is not possible, the quadratic formula can be used: x = (-b ± √(b^2 - 4ac)) / 2a.
Applications
Quadratic functions are used to model various real-world scenarios, such as:
- Projectile motion
- Optimization problems
- Electric circuits
- Physics and engineering
Quadratic Functions
- A quadratic function is a polynomial function of degree two, meaning the highest power of the variable is two.
Standard Form
- The standard form of a quadratic function is f(x) = ax^2 + bx + c.
- a, b, and c are constants, and a ≠0.
- x is the variable.
Graph
- The graph of a quadratic function is a parabola that opens upwards if a > 0 and downwards if a < 0.
- The vertex of the parabola is the lowest or highest point, depending on the direction of the opening.
Key Features
- The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two identical halves.
- The vertex is the lowest or highest point on the parabola.
- x-intercepts are the points where the parabola intersects the x-axis.
- y-intercept is the point where the parabola intersects the y-axis.
Solving Quadratic Equations
- Quadratic equations can be solved using factoring if the equation can be written in the form (x - r)(x - s) = 0.
- The solutions are x = r and x = s.
- If factoring is not possible, the quadratic formula can be used: x = (-b ± √(b^2 - 4ac)) / 2a.
Applications
- Quadratic functions are used to model projectile motion.
- Quadratic functions are used to model optimization problems.
- Quadratic functions are used to model electric circuits.
- Quadratic functions are used in physics and engineering.
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Description
Understand the definition and standard form of quadratic functions, and learn how to graph them. Explore the concepts of parabolas and vertices.
