MPM2D0 Final Exam

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THE BIOSPHERE

Alejandro Gonzálezにより

Trigonometry with The Unit Circle Cited and Colored

Widad Karsouにより

Standard Form

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Trigonometry with The Unit Circle Cited and Colored

Widad Karsouにより

MPM2D0 Final Exam

Quadratics

Methods of solving: factoring, completing the square, quadratic formula

Vertex Form

Factored Form

Standard form

Quadratic Formula

the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions.

Analytic Gemotry

Analytic Geometry connects algebra and geometry using coordinate systems. This involves studying lines, circles, and other geometric figures through algebraic equations.

Slope of a Line

Positive slope: Line rises from left to right Negative slope: Line falls from left to right Zero slope: Horizontal line Undefined slope: Vertical line

The slope describes the steepness and direction of a line

Midpoint Formula

Gives the coordinates of the midpoint between two points

Equation of a Circle

Special case: If h = 0, k = 0, this is the equation

Center: (h,k) Radius: r

Distance Formula

Important in geometric problems, finding the length of a line segment

Used to find the distance between two points in the coordinate plane

Equation of a Line

Slope-Point Form

Use this form when you know a point and the slope

Slope-Intercept Form

m = Slope b = y-intercept

Linear Systems

General Form

Point–slope form

Slope-intercept form

Polynomials

Polynomials are algebraic expressions involving a sum of powers of a variable with constant coefficients. Understanding polynomials is crucial for solving equations, graphing, and simplifying algebraic expressions. Key concepts include factoring, graphing, and operations on polynomials.

Degree of Polynomial

A polynomial with degree 1 is called a linear polynomial, degree 2 is quadratic, degree 3 is cubic, and so on.

The degree is the highest exponent of the variable in the polynomial.

Trigonometry

Trigonometry deals with the relationships between the angles and sides of triangles. The basic trigonometric functions—sine, cosine, and tangent—are used extensively in geometry, physics, and engineering.

Cosine Law

In trigonometry, the Cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

Sine law

In any plane triangle, the ratio of the sides is equal to the ratio of the sines of the angles opposite to those sides.

Pythagorean Theorem

Relates the sides of a right-angled triangle.

Formula

Soh Cah Toa

Basic Trigonometric Ratios:

Tangent: tan 𝜃

Opposite ------------ Adjacent

Cosine: cos 𝜃

Adjacent --------------- Hypotenuse

Sine: sin 𝜃

Opposite -------------- Hypotenuse