Algebra & Geometry Essentials Cheat Sheet

This cheat sheet covers fundamental concepts in algebra and geometry, including linear equations, quadratic functions, geometric properties, and key formulas for solving problems.

Core Principles

• Linear equations use y=mx+b form.
• Midpoint formula finds the center of a line segment.
• Distance formula calculates the length of a line segment.
• Circle equations depend on whether the center is at the origin.
• Trigonometric ratios (SOH CAH TOA) apply to right triangles.
• Quadratic equations can be solved using the quadratic formula.
• Properties of quadrilaterals (parallelograms, rectangles, squares, rhombuses, trapezoids, kites) define their sides and angles.
• Factoring is a key method for solving quadratic equations.

Action Steps

• For linear equations, use substitution or elimination to solve for variables.
• Calculate midpoint by averaging x and y coordinates.
• Use distance formula for segment length.
• Identify circle center and radius from its equation.
• Apply Sine/Cosine Law for non-right triangles.
• Use SOH CAH TOA for right triangle trig ratios.
• Factor quadratics by finding two numbers that multiply to 'c' and add to 'b' (for a=1).
• Use the quadratic formula when factoring is difficult.
• Convert standard quadratic form to vertex form by completing the square.
• Analyze quadrilateral properties to identify shape and solve for unknown angles/sides.

Formulas

• Midpoint: $M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$
• Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Circle (center at origin): $x^2 + y^2 = r^2$
• Circle (center at (h,k)): $(x-h)^2 + (y-k)^2 = r^2$
• Sine Law: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Cosine Law: $c^2 = a^2 + b^2 - 2bc \cos A$
• SOH: $\sin = \frac{opposite}{hypotenuse}$
• CAH: $\cos = \frac{adjacent}{hypotenuse}$
• TOA: $\tan = \frac{opposite}{adjacent}$
• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
• Standard Quadratic Form: $y = ax^2 + bx + c$
• Vertex Form: $y = a(x-h)^2 + k$

Key Terms

• Slope: Rate of change of a line; rise over run ($m = \frac{y_2-y_1}{x_2-x_1}$).
• Perpendicular Bisector: A line that intersects another line segment at its midpoint and is perpendicular to it.
• Centroid: The intersection of the medians of a triangle; located 1/3 of the distance from the midpoint of a side to the opposite vertex.
• Median: A line segment joining a vertex to the midpoint of the opposite side.
• Vertex: The highest or lowest point on a parabola.
• Axis of Symmetry: A line that divides a parabola into two mirror images.

Pro Tips

• Remember that 'a' in $y=ax^2+bx+c$ determines if the parabola opens up (a>0) or down (a<0).
• A larger absolute value of 'a' makes the parabola narrower.
• The 'b' term shifts the parabola left/right, and 'c' shifts it up/down.
• Always factor out the Greatest Common Factor (GCF) first when factoring.
• Lines with the same slope are parallel; lines with negative reciprocal slopes are perpendicular.
• The centroid is always 2/3 of the distance from the vertex to the midpoint of the opposite side.

Pitfalls to Avoid

• Confusing substitution and elimination methods.
• Forgetting to square terms in the distance formula.
• Applying trig ratios to non-right triangles.
• Errors in sign when using the quadratic formula.
• Incorrectly identifying quadrilateral properties.
• Mistakes in completing the square for vertex form.

Real World Examples

• Calculating the distance between two cities on a map.: Using the distance formula.
• Determining the trajectory of a thrown ball.: Modeling with quadratic functions.
• Architectural designs involving angles and lengths.: Applying geometric principles and trigonometric laws.

Timeline

• Ancient Greece: Development of Euclidean geometry, including theorems on triangles and quadrilaterals.
• 17th Century: René Descartes and Pierre de Fermat develop analytic geometry, linking algebra and geometry.
• 17th-18th Century: Development of calculus by Newton and Leibniz, providing tools for analyzing curves and rates of change.
• 19th Century: Non-Euclidean geometries explored, expanding the understanding of space.
• 20th Century - Present: Continued application and expansion of algebraic and geometric concepts in fields like computer graphics, physics, and engineering.

People

• René Descartes: Pioneer of analytic geometry, linking algebra and geometry.
• Pierre de Fermat: Contributed to analytic geometry and number theory.
• Pythagoras: Associated with the Pythagorean theorem ($a^2+b^2=c^2$) for right triangles.

Quiz

• What is the slope of a line perpendicular to $y = 2x + 3$?: -1/2
• Which formula calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$?: Distance formula
• For the quadratic equation $y = -x^2 + 4x - 1$, does the parabola open upwards or downwards?: Downwards