EOC Review Notes

Function Types from Tables

1. Table 1:

   • X values increase by 1.
   • F(x) increases by 3 each time.
   • Conclusion: Linear function (consistent first difference).

2. Table 2:

   • X values increase by 1.
   • Multiplication pattern by 4.
   • Conclusion: Exponential function (consistent multiplication factor).

Evaluating Functions

• Example: Evaluate f(-3) for a given quadratic expression.
• Substitute -3 for x in the expression.
• Simplify using arithmetic and properties of exponents.
• Result: f(-3) = -8.

Quadratic Functions

• Key Features:
  • Vertex: Point where the parabola turns, either maximum or minimum (e.g., (-1, 4)).
  • Axis of Symmetry: x = value of x-coordinate of the vertex.
  • Domain: All real numbers.
  • Range: Based on vertex up or down (e.g., y ≤ 4).
  • End Behavior: Parabolic arms pointing down (y → -∞).
  • Intervals: Increasing up to vertex, decreasing after.

Solving Systems of Equations

• Method: Elimination by aligning coefficients and eliminating variables.
• Solve for remaining variable and substitute back.
• Example: Solution (x, y) = (2, 2).

Simplifying Expressions with Exponents

• Negative Exponents: Move base across the fraction bar to change sign.
• Simplify by factoring and reducing common terms.
• Example: x * y^5 / 3 after simplification.*

Radical Expressions

• Simplification with Cube Roots:
  • Factor into perfect cubes.
  • Extract and simplify cube roots.
  • Example: 4 * cube root(40) simplified to 16 * cube root(20).

Graphing Linear Equations

• Slope-Intercept Form: y = mx + b.
• Steps:
  • Plot y-intercept.
  • Use slope to find another point.
  • Draw a line through points.

Comparing Linear Functions

• Slope: Rate of change y/x.
• Y-intercept: Value at x = 0.
• Comparison: Across different functions for greater values.

Writing Inequalities from Graphs

• Linear Inequality: Shading direction determines inequality.
• Dashed Line: No equal bar in inequality (e.g., y > -x - 2).

Absolute Value Equations

• Method: Set up two equations, one equal to positive and another to negative of the absolute value.
• Solve both for the variable.

Simple Interest

• Formula: A = P(1 + rt).
• Calculate using initial principal, rate, and time.

System of Equations from Word Problems

• Setup: Use information to form equations.
• Elimination: Adjust equations to eliminate variables and solve.

Parabolic Equations

• Vertex Form: y = a(x - h)^2 + k.
• Use known points to find 'a' and convert to standard form.

Maximizing Profits

• Quadratic Function: Use formula -b/2a to find vertex.
• Application: Determines optimal pricing.

Perpendicular Line Equations

• Slope: Opposite reciprocal of original line's slope.
• Point-Slope Form: y - y1 = m(x - x1).

End Behavior of Exponential Functions

• Graph Analysis: Determine limits as x approaches ±∞.
• Behavior: Function may approach zero or diverge based on coefficient and base.

These notes summarize key concepts and example problems from the EOC review lecture, providing guidance in recognizing and solving different types of mathematical problems typically encountered in end-of-course assessments.