Linear Equations: Slope, Intercept, and Forms

• Linear equations involve variables with a maximum power of one.

Slope of a Line

• The slope (m) of a line measures its steepness and direction.
• Slope is calculated as the change in y divided by the change in x (rise over run).
• The formula for slope is: m = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are two points on the line.
• A positive slope indicates an increasing line (from left to right).
• A negative slope indicates a decreasing line (from left to right).
• A zero slope indicates a horizontal line.
• An undefined slope indicates a vertical line.

Y-Intercept

• The y-intercept (b) is the point where the line crosses the y-axis.
• It is the value of y when x = 0.
• In the slope-intercept form of a linear equation (y = mx + b), 'b' represents the y-intercept.

Slope-Intercept Form

• The slope-intercept form of a linear equation is y = mx + b.
• 'm' represents the slope of the line.
• 'b' represents the y-intercept of the line.

Point-Slope Form

• The point-slope form of a linear equation is y - y₁ = m(x - x₁).
• m is the slope
• (x₁, y₁) is a known point on the line.
• This form is useful for finding the equation of a line when you know the slope and a point on the line.

Finding the Intersection of Two Lines

• The intersection point is where two lines cross each other.
• At the intersection point, the x and y values are the same for both lines.
• Several methods pinpoint the intersection:
  • Graphing: Plot both lines and visually identify the intersection point.
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination (or Addition/Subtraction): Manipulate the equations so that one variable has opposite coefficients, then add the equations together to eliminate that variable.

Simultaneous Equations

• Simultaneous equations (also called a system of equations) involve two or more equations with the same variables.
• The solution to a system of equations provides values for the variables that satisfy all equations simultaneously.
• Methods for solving:
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation(s).
  • Elimination: Manipulate the equations to eliminate one variable by adding or subtracting the equations.
  • Graphing: Graph each equation and find the point(s) of intersection.

Piecewise Graphs

• A piecewise function is defined by multiple sub-functions, each applicable over a specific interval of the function's domain.
• The graph of a piecewise function comprises the graphs of its sub-functions across their defined intervals.
• Piecewise functions are useful for modeling situations where variable relationships change based on the input.
• When graphing, close attention should be paid to interval endpoints, using open circles to denote exclusion and closed circles to denote inclusion.

Step Graphs

• A step graph represents a piecewise graph where sub-functions remain constant over each interval, resulting in horizontal steps.
• Step graphs commonly model situations where the output value jumps to a new level at specific intervals.
• The greatest integer function, f(x) = ⌊x⌋, exemplifies a step function by returning the largest integer less than or equal to x.
• The ceiling function, f(x) = ⌈x⌉, also constitutes a step function, yielding the smallest integer greater than or equal to x.
• Step graphs may use open or closed circles at each step's beginning and end to clarify endpoint inclusion.