Overview
This lecture reviews the core concepts of linear functions, including their forms, graphing, properties, and key applications in mathematics.
Prerequisites for Linear Functions
- Must be able to write numbers as fractions.
- Can evaluate simple linear functions for given values of m, x, and b.
- Should be able to distribute and combine like terms.
- Able to solve linear equations by isolating variables.
Forms and Graphs of Linear Functions
- The slope-intercept form is ( f(x) = mx + b ), where m is slope, b is y-intercept.
- The identity function is when ( m = 1 ), ( b = 1 ).
- To graph, start at the y-intercept (b), use the slope (rise over run) to plot another point, and draw a line.
- If ( m = 0 ), the line is horizontal; if ( m > 0 ), the line increases; if ( m < 0 ), the line decreases.
Domain, Range, and Intercepts
- The domain of any linear function is all real numbers.
- The range is all real numbers unless ( m = 0 ), then the range is a single value.
- The y-intercept is b; the x-intercept can be found by setting ( f(x) = 0 ) and solving for x.
- There is always one y-intercept, and the number of x-intercepts depends on m and b.
Special Features and Properties
- Linear functions do not have asymptotes.
- They are always continuous.
- Monotonicity depends on slope: increasing if ( m > 0 ), decreasing if ( m < 0 ), constant if ( m = 0 ).
- Symmetry: function is even if ( m = 0 ), odd if ( b = 0 ).
Finding Equations from Points
- To find the equation, first determine the slope using two points.
- Use the point-slope form or set up a system of equations to solve for m and b.
Applications and Interpretation
- m (slope) is the rate of change of output per unit input.
- b (y-intercept) is the output when input is zero (initial value).
- Units of b match output units; units of m are output units divided by input units.
Key Terms & Definitions
- Linear function — A function in the form ( f(x) = mx + b ).
- Slope (m) — The change in output per unit change in input.
- Y-intercept (b) — The value where the graph crosses the y-axis.
- Domain — All possible input values for the function.
- Range — All possible output values for the function.
- Monotonicity — Whether a function is always increasing, decreasing, or constant.
- Even function — Symmetric about the y-axis.
- Odd function — Symmetric about the origin.
Action Items / Next Steps
- Practice graphing linear functions in slope-intercept form.
- Review solving linear equations for x- and y-intercepts.
- Remember the meaning and units for slope and y-intercept in applications.