Linear Equations & Graphs Cheat Sheet

Master the fundamentals of linear equations and their graphical representations to solve problems and understand relationships between variables.

Core Principles

• Linear equations represent a straight line.
• The slope indicates the steepness and direction of a line.
• The y-intercept is where the line crosses the y-axis.
• Quadratic equations form a parabola when graphed.
• Real-world problems can be modeled using linear equations.

Action Steps

• Identify variables (x and y).
• Determine the slope (m) and y-intercept (b).
• Plot the y-intercept on the graph.
• Use the slope to find additional points.
• Draw a straight line through the points.
• For quadratics, find roots using factoring or the quadratic formula.
• Substitute values to check solutions.

Formulas

• Slope-intercept form: $y = mx + b$
• Point-slope form: $y - y_1 = m(x - x_1)$
• Standard form: $Ax + By = C$
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Key Terms

• Linear Equation: An equation whose graph is a straight line.
• Slope (m): The measure of the steepness of a line; rise over run.
• Y-intercept (b): The point where a line crosses the y-axis (x=0).
• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$ where a ≠ 0.
• Parabola: The U-shaped graph of a quadratic equation.

Real World Examples

• Cost of a taxi ride: Base fare (y-intercept) plus cost per mile (slope).
• Distance traveled at constant speed: Distance = Speed (slope) * Time + Initial Distance (y-intercept).
• Projectile motion: Path of a thrown object follows a parabolic (quadratic) curve.