Stem-and-Leaf Plots, Histograms, Systems of Equations

• The test consists of 15 questions.

Displaying Data in Stem-and-Leaf Plots and Histograms

• Stem-and-leaf plots can be used to find mean, median, mode, and range.
• Mean: the average of all data values.
• Median: the middle number.
• Mode: the most frequently occurring value.
• Range: the difference between the highest and lowest values.
• Relative Frequency can be expressed as (number of times value occurs) / (Total number of data values)

Solving and Classifying Special Systems of Equations

• Inconsistent systems have no common solution, resulting in parallel lines when graphed.
• Algebraically, an inconsistent system yields a statement that is never true.
• Consistent and dependent systems result in the same line when graphed.
• The equations are identical when converted to slope-intercept form
• Algebraically, this system yields a statement that is always true.
• Consistent and independent systems have one common solution.
• Graphically, there is one intersection.
• Algebraically, the system yields one solution, represented as a specific point (x,y).

Mutually Exclusive and Inclusive Events

• Mutually exclusive events cannot occur at the same time.
• P(A or B) = P(A) + P(B) for mutually exclusive events.
• Inclusive events can both occur at the same time.
• P(A or B) = P(A) + P(B) - P(A and B) for inclusive events.

Adding and Subtracting Radical Expressions

• Only like radicals can be combined.
• Like radicals have the same index and radicand.

Solving Absolute Value Equations

• Isolate the absolute value expression before solving.
• If |x| = b, then x = b OR x = -b.
• Word problem pattern: |𝑥 − (𝑠𝑒𝑡 𝑜𝑟 𝑔𝑖𝑣𝑒𝑛 𝑣𝑎𝑙𝑢𝑒)| =± 𝑎𝑚𝑜𝑢𝑛𝑡

Multiplying Radical Expressions

• √𝑎𝑏 = √𝑎√𝑏 to be true, a and b must be greater than zero.

Solving Inequalities with Variables on Both Sides

• Solve inequalities like equations.
• A contradiction is an inequality that is never true.
• Ex: solving results in a statement such as 2 < -3.
• An identity is an inequality that is always true.
• Ex: solving results in -2 < 3.

Solve Multi-Step Compound Inequalities

• Uses inequalities with AND or OR.
• Solve using inverse operations, similar to other inequalities.
• Pay special attention to multiplying or dividing by a negative number.

Factoring Special Products

• Perfect-Square Trinomials: a² + 2ab + b² factors to (a + b)².
• a² - 2ab + b² factors to (a - b)².
• Difference of two squares: (a² - b²) factors to (a + b) (a - b).
• Look for patterns to factor quicker
• Focus on squares and the middle term being 2(a)(b).
• Remember to factor out any GCF before looking for patterns

Identifying Quadratic Functions

• Standard form: f(x) = ax² + bx + c, where a, b, and c are real numbers and a ≠ 0.
• The graph is a parabola, U-shaped.
• If a < 0, the parabola opens downward.
• If a > 0, the parabola opens upward.

Solving Problems Using the Pythagorean Theorem

• Relates the sides of a right triangle: a² + b² = c², where c is the hypotenuse.
• 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

• Use when there are 4 terms.
• Rearrange terms if needed to factor out a common factor from each set of two terms.
• Always factor out the GCF of all terms first.

Multiplying and Dividing Rational Expressions

• Simplify all exponents so that you have no zero or negative powers
• With all rational expressions, excluded values must be determined

Identifying Characteristics of Quadratic Functions

• Vertex: the highest or lowest point on the parabola.
• The minimum is the least possible y-value of the function.
• The maximum is the greatest possible y-value of the function.
• Zero, Root, x-intercept: where the function crosses the x-axis.
• Axis of Symmetry: a vertical line that divides the parabola into two mirror images and passes through the vertex.
• Formula for axis of symmetry: x = -b / 2a