Stem-and-Leaf Plots, Histograms and Equation Systems

• Test 18 will have 15 questions

Displaying Data in Stem-and-Leaf Plots and Histograms

• Stem and leaf plots can find mean, median, mode, and range.
• Mean is the average and calculated by adding all data values and dividing by the number of values.
• Median is the middle number
• Mode is the most occurring value
• Range is the highest value minus the lowest value
• Relative frequency is calculated by dividing the number of times a value occurs by the total number of data values

Solving and Classifying Special Systems of Equations

• Inconsistent systems have no common solution, the lines are parallel when graphed, and solving algebraically results in a statement that is never true. (e.g., 3 = 7)
• Consistent and Dependent systems result in the same line whether graphed or put in slope-intercept form, and solving algebraically results in a statement that is always true. (e.g., 4 = 4)
• Consistent and Independent system: When graphed, one common solution means one intersection, and solving algebraically yields one solution (a specific point (x,y))

Mutually Exclusive and Inclusive Events

• Mutually Exclusive Events are events that cannot both happen at the same time, found with P(A or B) = P(A) + P(B)
• Inclusive Events are events where both can occur at the same time, found with P(A or B) = P(A) + P(B) - P(A and B)

Adding and Subtracting Radical Expressions

• Radicals can only combine like radicals
• Like radicals have the same index and radicand

Solving Absolute Value Equations

• You must isolate absolute value first: |x| = b, then x = b OR x = -b
• Word problem pattern is |𝑥 − (𝑠𝑒𝑡 𝑜𝑟 𝑔𝑖𝑣𝑒𝑛 𝑣𝑎𝑙𝑢𝑒)| =± 𝑎𝑚𝑜𝑢𝑛𝑡

Multiplying Radical Expressions

• √𝑎𝑏 = √𝑎 √𝑏, where a and b must be greater than zero

Solving Inequalities with Variables on Both Sides

• Solve the inequality as an equation
• Contradiction: An inequality that is never true (e.g., 2 < -3)
• Identity: An inequality that is always true (e.g., -2 < 3)

Solve Multi-Step Compound Inequalities

• Contains Inequalities with AND or OR statements
• Use inverse operations to solve the inequalities like other inequalities
• Pay attention when multiplying or dividing by a negative

Factoring Special Products

• Perfect-Square Trinomials: a² + 2ab + b² factors as (a + b)² and a² - 2ab + b² factors as (a - b)²
• Difference of two squares: (a² + b²) factors as (a + b) (a - b)
• In a perfect square trinomial, the first and last terms are perfect squares, and the middle term is 2(a)(b)
• Difference of two squares have two terms that are perfect squares separated by subtraction
• Factor out any GCF before looking for patterns

Identifying Quadratic Functions

• The standard form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0
• The graph is a parabola, which is U-shaped
• If a < 0, the parabola opens downward
• If a > 0, the parabola opens upwards

Solving Problems Using the Pythagorean Theorem

• The sum of squares of the lengths of the legs equals the square of the hypotenuse: a² + b² = c²
• Pythagorean Triple: A set of three nonzero whole numbers that satisfy the Pythagorean Theorem

Calculating the Midpoint and Length of a Segment

• Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)

Factoring Polynomials by Grouping

• Used when there are 4 terms
• You may need to rearrange terms in order to factor out a common factor of each set of two terms
• Factor out any GCF of all terms first (Be careful not to lose the GCF in your answer)

Multiplying and Dividing Rational Expressions

• Simplify all exponents so that there are no zero or negative powers
• With all rational expressions, make sure that you list any excluded values

Identifying Characteristics of Quadratic Functions

• Vertex: The highest or lowest point of the parabola graph
• Minimum: Least possible value of the function
• Maximum: Greatest possible value of the function
• The minimum or maximum is the y-value of the vertex
• Zero/root/x-intercept: different names for the x value where the function crosses the x axis
• Axis of symmetry: Vertical line that divides the graph into two mirror images, goes through the vertex
• Formula for axis of symmetry: x = -b / 2a