Quadratic Equations and Parabolas Quiz

Study Notes

Quadratic Equations

Defined in standard form as f(x) = ax² + bx + c, where a ≠ 0.
Quadratic term is represented by x²; linear term is represented by x; and constant term is simply a number.

Parabolas and Symmetry

The graph of a quadratic function is called a parabola.
The axis of symmetry is a vertical line that passes through the vertex of the parabola, with points mirrored on either side.

Modeling and Factoring

To model data with a quadratic function: convert to standard form, solve a system of three equations, and write the function.
Factoring methods include GCF, difference of squares, guess and check, and the British method to find roots.

Special Polynomial Forms

Perfect square trinomials indicate specific relationships among coefficients: B = (√a) + (√c) x 2.
Completing the square involves isolating terms, adding squares, and factoring the resulting trinomial.

Solving Quadratic Equations

Roots can be found through factoring and applying the Zero Product Property: if ab = 0, then a = 0 or b = 0.
The quadratic formula (x = (-b ± √(b² - 4ac)) / 2a) finds roots directly from any quadratic equation.

Discriminant Analysis

The discriminant (D = b² - 4ac) indicates the nature of the roots:
- Positive D means two real x-intercepts.
- Negative D means no real x-intercepts.
- Zero D means one real x-intercept, leading to the vertex lying on the x-axis.

Complex Numbers

Imaginary unit "i" represents √-1; powers of i cycle: i, -1, -i, 1.
Complex numbers are expressed as a + bi, where a (real part) and b (imaginary part) are real numbers.

Complex Number Operations

To add or subtract complex numbers, combine corresponding real and imaginary parts.
For multiplication, use FOIL method; for division, multiply by the conjugate of the denominator.

Absolute Value and Inverses

The absolute value of a complex number represents its distance from the origin in the complex plane: |a + bi| = √(a² + b²).
The additive inverse of a complex number is its opposite.

Graphing and Representation

Complex numbers can be plotted on a plane, with real numbers along the horizontal axis and imaginary numbers along the vertical axis.
Perfect square trinomials can be easily recognized as factors are identical.

Quadratic Forms

Standard form: y = ax² + bx + c; vertex calculated as (-b/2a, f[-b/2a]).
Vertex form: y = a(x - h)² + k; vertex is (h, k), with the axis of symmetry determined by x = h.

Behavior of Parabolas

If a > 0, the parabola opens upwards and has a minimum point at its vertex.
The y-intercept can be found by evaluating when x = 0, while x-intercepts are found by solving y = 0.

Description

Test your knowledge on quadratic equations, their standard forms, and the properties of parabolas. This quiz covers modeling, factoring techniques, and methods for solving quadratic equations. Perfect for students looking to strengthen their understanding of this essential mathematical concept.