Algebra 2 Midterm Study Guide

Math Midterm Study Guide

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Unit 0: Basics of Statistics and Data Representation

Z Score

• A measure of how far a data point is from the mean, expressed in standard deviations.
• Formula: Z = (X - mean) / standard deviation
• Interpretation:
  • Z > 0: Data point is above the mean.
  • Z < 0: Data point is below the mean.

Mean and Standard Deviation

• Mean: The average of a data set.
  • Formula: Mean = (Sum of all values) / (Number of values)
• Standard Deviation: A measure of how spread out the data points are around the mean.

Skewed Data

• Right-Skewed (Positive): Mean > Median, tail on the right.
• Left-Skewed (Negative): Mean < Median, tail on the left.

Marginal Frequency Tables

• Summarize frequencies of two categorical variables.
• Marginal frequencies are the row and column totals in the table.

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Unit 1: Linear Functions and Systems

Interval Notation

• Closed Interval: Includes the endpoints, written as [a, b].
• Open Interval: Excludes the endpoints, written as (a, b).
• Example:
  • x ≥ 2: [2, ∞)
  • 0 < x < 3: (0, 3)

Slope-Intercept Form

• Equation: y = mx + b
  • m: slope (rise/run)
  • b: y-intercept (value of y when x = 0)

Equations Passing Through Points

1. Find the slope: m = (y2 - y1) / (x2 - x1)
2. Use point-slope form: y - y1 = m(x - x1)

Reflections and Transformations

• Horizontal shift: f(x-h) shifts right by h.
• Vertical shift: f(x) + k shifts up by k.
• Reflection over the x-axis: -f(x).
• Reflection over the y-axis: f(-x).
• Stretch/Compress: Multiply f(x) by a value greater or less than 1.

Function Types

• Linear: y = mx + b
• Quadratic: y = ax² + bx + c
• Absolute Value: y = a|x-h| + k

Solving Systems of Equations

• Substitution: Solve one equation for one variable and substitute into the other.
• Elimination: Add or subtract equations to eliminate a variable.
• Graphing: Plot both equations and find their intersection point.

Scatter Plots

• Analyze the relationship between two variables.
• Identify positive, negative, or no correlation.

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Unit 2: Quadratics and Transformations

Transformations

• Parent functions can be shifted, stretched, compressed, or reflected.
• Example:
  • Horizontal shift: f(x-h)
  • Vertical shift: f(x) + k
  • Reflection: -f(x)

Domain and Range

• Domain: All possible x-values.
• Range: All possible y-values.

Vertex and Axis of Symmetry

• Vertex: The turning point of a parabola.
  • Formula for x: -b / (2a)
  • Plug x into the equation to find the y-coordinate.
• Axis of Symmetry: The vertical line that passes through the vertex.

Match Equation to Graph

• Recognize how changes in the equation affect the graph (shifts, reflections, and stretches).

Quadratic Forms

• Vertex Form: y = a(x-h)² + k
• Standard Form: y = ax² + bx + c
• Intercept Form: y = a(x-p)(x-q)

Minimum/Maximum Value

• For y = ax² + bx + c:
  • If a > 0: Minimum value at the vertex.
  • If a < 0: Maximum value at the vertex.

Average Rate of Change

• Formula: (f(x2) - f(x1)) / (x2 - x1)
• Represents the slope between two points on a curve.

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Unit 3: Quadratic Solutions

Discriminant

• Formula: b² - 4ac
  • Positive: Two real solutions.
  • Zero: One real solution.
  • Negative: Two complex solutions.

Solving Methods

• Factoring: Break the equation into two binomials and solve.
• Completing the Square: Rearrange into the form (x-h)² = k.
• Quadratic Formula:
  • x = [-b ± √(b² - 4ac)] / (2a)

Imaginary Numbers

• i = √(-1)
• Properties:
  • i² = -1
  • i³ = -i
  • i⁴ = 1

Vertex Form

• Rewrite a quadratic in standard form to vertex form: y = a(x-h)² + k

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Unit 4: Polynomials

Graphing Polynomials

• End Behavior: Determined by the degree and leading coefficient.
• Roots: Where the graph crosses or touches the x-axis.

Polynomial Operations

• Addition/Subtraction: Combine like terms.
• Multiplication: Distribute or use FOIL.
• Division: Use synthetic or long division.

Factoring Methods

• Slip and Slide: Multiply the leading coefficient and constant, factor, then divide back.
• Grouping: Group terms and factor common terms.
• Special Factoring:
  • Difference of Squares: a² - b² = (a-b)(a+b)
  • Perfect Square Trinomial: a² + 2ab + b² = (a+b)²

Factoring Higher Powers

• Factor by grouping or recognizing patterns.

Solving Polynomial Equations

• Steps:
  1. Factor the polynomial.
  2. Set each factor equal to zero.
  3. Solve for x.

Fundamental Theorem of Algebra

• A polynomial of degree n has exactly n roots (real or complex).

Pascal’s Triangle

• A tool for binomial expansions:
  • Row numbers correspond to the powers of the binomial.
  • Example: (a+b)² = 1a² + 2ab + 1b²