EOC Review Lecture Notes

Function Identification from Tables

• Table 1: X values increase by 1, f(x) values increase by 3.
  • Conclusion: Linear function due to constant first differences.
• Table 2: X values increase by 1, f(x) values increase by multiplying by 4.
  • Conclusion: Exponential function due to repeated multiplication (times 4).

Evaluating Functions

• Example: Evaluate f(-3) for a given quadratic function.
  • Substitute x = -3 into f(x).
  • Calculation: f(-3) = -18 + 9 + 1 = -8.

Key Features of Quadratic Functions

• Vertex: The highest or lowest point (e.g., (-1, 4)).
• Axis of Symmetry: x = -1 (from vertex x-coordinate).
• Domain and Range:
  • Domain: All real numbers.
  • Range: y ≤ 4 (maximum value).
• End Behavior: As x → ±∞, y → -∞ (both ends of the parabola point down).
• Increasing/Decreasing Intervals:
  • Increasing: x < -1.
  • Decreasing: x > -1.

Solving Systems of Equations

• Method Used: Elimination.
  • Example: Eliminate variables by multiplying and adding equations.
• Example Solution: x = 2, y = 2.

Simplifying Expressions with Exponents

• Negative Exponents: Move terms between numerator and denominator to make exponents positive.
  • Example: Simplify to x * y^5 / 3 after adjustments.

Solving Radical Expressions

• Cube Roots: Identify perfect cubes (like 8, 27) for simplification.
• Simplifying: Combine terms under a single radical (e.g., 16 * cube root of 20).

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b.
  • Example: Slope = 3/4, y-intercept = -2.
  • Plot points using rise over run.

Comparing Linear Functions

• Slope Comparison: Determine greater slope between functions.
• Intercepts:
  • Y-Intercept: Found by setting x = 0.
  • X-Intercept: Found by setting y = 0 (solve for x).

Writing Inequalities from Graphs

• Identifying Inequality: Based on line shading (above = ">", dashed line = no "=").
• Example: y > -x - 2.

Solving Absolute Value Equations

• Steps: Isolate absolute value, solve two resulting equations.
• Example Solutions: x = 0 or x = -1.

Calculating Simple Interest

• Formula: A = P(1 + rt).
• Example: Calculate total after 10 years for given P and r.

Solving Word Problems with Systems

• Scenario: Solve cost of shirts and pants using elimination.
• Solution: x = $16.66 (shirts), y = $16.67 (pants).

Writing Equation of a Parabola

• Vertex Form: y = a(x - h)^2 + k.
• Convert to Standard Form: Expand and combine like terms.
• Verification: Plug in points to verify equation accuracy.

Maximizing Profit in Quadratics

• Vertex Calculation: Use -b/(2a) for maximum price calculation.
• Example: Maximize profit with $3.50 ticket price.

Writing Perpendicular Line Equations

• Identify Perpendicular Slope: Negative reciprocal of given slope.
• Point-Slope Form: Convert to slope-intercept form.

Describing End Behavior of Functions

• Exponential Function: End behavior as x approaches ±∞.
• Y-Value Behavior: Describe increase/decrease towards zero.