Algebra Basics Quiz

Algebra

• Equations and Inequalities:
  • Linear equations: ax + by = c, where a, b, and c are constants
  • Quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
  • Solving systems of linear equations using substitution and elimination methods
• Functions:
  • Domain and range of a function
  • Composition of functions
  • Inverse functions
• Graphing:
  • Graphing linear and quadratic equations
  • Identifying key features of graphs (x-intercepts, y-intercepts, vertex, etc.)

Trigonometry

• Angles and Triangles:
  • Measuring angles in degrees and radians
  • Properties of triangles (Pythagorean identity, sum of angles)
• Trigonometric Functions:
  • Sine, cosine, and tangent functions
  • Graphs and identities of trigonometric functions
  • Solving trigonometric equations
• Applications:
  • Modeling periodic phenomena (sound waves, light waves)
  • Solving problems involving right triangles

Geometry

• Points, Lines, and Planes:
  • Properties of points, lines, and planes (midpoint, distance, slope)
  • Relationships between points, lines, and planes (parallel, perpendicular, intersecting)
• Angles and Measurement:
  • Measuring angles and arcs in degrees and radians
  • Properties of angles (complementary, supplementary, corresponding)
• Shapes and Solids:
  • Properties of 2D and 3D shapes (perimeter, area, volume)
  • Calculating perimeter, area, and volume of various shapes

Transformations

• Translations:
  • Translating shapes and points on a coordinate plane
  • Notation and formulas for translations
• Rotations:
  • Rotating shapes and points on a coordinate plane
  • Notation and formulas for rotations
• Reflections:
  • Reflecting shapes and points on a coordinate plane
  • Notation and formulas for reflections
• Combinations:
  • Combining multiple transformations to create a new transformation
  • Notation and formulas for combined transformations

Algebra

• Linear equations are represented as ax + by = c, where a, b, and c are constants.
• Quadratic equations are represented as ax^2 + bx + c = 0, where a, b, and c are constants.
• Systems of linear equations can be solved using substitution and elimination methods.

Functions

• The domain of a function is the set of input values, while the range is the set of output values.
• Composition of functions involves combining two or more functions.
• Inverse functions are functions that "undo" the operation of another function.

Graphing

• Linear and quadratic equations can be graphed on a coordinate plane.
• Key features of graphs include x-intercepts, y-intercepts, and vertex.

Trigonometry

Angles and Triangles

• Angles can be measured in degrees and radians.
• Properties of triangles include the Pythagorean identity and the sum of angles.

Trigonometric Functions

• Sine, cosine, and tangent functions relate to the ratios of sides in triangles.
• Graphs of trigonometric functions have unique identities and properties.
• Trigonometric equations can be solved using algebraic methods.

Applications

• Trigonometry can be used to model periodic phenomena, such as sound waves and light waves.
• Right triangles can be used to solve problems involving heights and distances.

Geometry

Points, Lines, and Planes

• Points have no size, lines have no thickness, and planes have no depth.
• Properties of points, lines, and planes include midpoint, distance, and slope.
• Relationships between points, lines, and planes include parallel, perpendicular, and intersecting.

Angles and Measurement

• Angles can be measured in degrees and radians.
• Properties of angles include complementary, supplementary, and corresponding.

Shapes and Solids

• Properties of 2D shapes include perimeter and area.
• Properties of 3D shapes include volume in addition to perimeter and area.
• Formulas exist for calculating perimeter, area, and volume of various shapes.

Transformations

Translations

• Translations involve moving shapes and points on a coordinate plane.
• Notation and formulas exist for translations, such as (x, y) → (x + h, y + k).

Rotations

• Rotations involve turning shapes and points on a coordinate plane.
• Notation and formulas exist for rotations, such as (x, y) → (x cos θ - y sin θ, x sin θ + y cos θ).

Reflections

• Reflections involve flipping shapes and points on a coordinate plane.
• Notation and formulas exist for reflections, such as (x, y) → (x, -y).

Combinations

• Multiple transformations can be combined to create a new transformation.
• Notation and formulas exist for combined transformations, such as a translation followed by a rotation.

Algebra

• Variables represent unknown values, while constants are fixed values
• Algebraic expressions combine variables, constants, and mathematical operations
• Equations and inequalities are statements that two expressions are equal or not equal
• Linear equations have the highest power of the variable(s) as 1, e.g. 2x + 3 = 5
• Quadratic equations have the highest power of the variable(s) as 2, e.g. x^2 + 4x + 4 = 0

Trigonometry

• Angles are measures of rotation, measured in degrees or radians
• Trigonometric ratios relate sides of a right-angled triangle, including sine, cosine, and tangent
• Sine, cosine, and tangent formulas:
  • sin(A) = opposite side / hypotenuse
  • cos(A) = adjacent side / hypotenuse
  • tan(A) = opposite side / adjacent side
• Inverse trigonometric functions (arcsin, arccos, arctan) find the angle given the ratio

Geometry

• Points are locations in space, represented by coordinates
• Lines are sets of points extending infinitely in two directions
• Planes are flat surfaces extending infinitely in all directions
• Angles are measures of rotation, measured in degrees or radians
• Triangles are polygons with three sides and three angles, including right-angled triangles

Transformations

• Translation: moving a shape by a certain distance in a certain direction
• Rotation: rotating a shape by a certain angle around a certain point
• Reflection: flipping a shape over a certain line or axis
• Enlargement: scaling a shape up or down by a certain factor

Pythagoras Theorem

• a^2 + b^2 = c^2, where a and b are the lengths of the legs of a right-angled triangle, and c is the length of the hypotenuse
• Used to find the length of the hypotenuse, or the length of one leg given the other leg and the hypotenuse

Angles of Elevation and Depression

• Angle of elevation: the angle between the line of sight and the horizontal, when looking up at an object
• Angle of depression: the angle between the line of sight and the horizontal, when looking down at an object
• Used to find the height of an object, or the distance to an object, given the angle and other measurements

Graphs

• Cartesian plane: a grid of points with x and y axes, used to plot graphs
• Coordinates: points on the graph, represented by an ordered pair (x, y)
• Linear graphs: graphs of linear equations, which are straight lines
• Quadratic graphs: graphs of quadratic equations, which are parabolas

Quadratic Graphs

• Parabolas: U-shaped curves that open up or down
• Vertex: the lowest or highest point of the parabola
• Axis of symmetry: the vertical line that passes through the vertex
• x-intercepts: the points where the graph crosses the x-axis, found by solving the quadratic equation