Linear Functions, Graphs, and Systems of Equations: Precalculus Study Notes

Linear Functions and Their Graphs

Definition and General Form

Linear functions are algebraic expressions that describe straight lines on the Cartesian plane. The general form of a linear equation is:

General Form:

$y = mx + b$

• m is the slope (rate of change), representing how much y changes for each unit increase in x.

• b is the y-intercept, the value of y when x = 0.

Slope and Y-Intercept

• Slope (m): Determined by the units risen on the y-axis and the units run on the x-axis. It measures the steepness and direction of the line.

• Y-intercept (b): The point where the line crosses the y-axis.

Example: For $y = 2x + 3$, the slope is 2 and the y-intercept is 3.

Graphing Linear Equations

• To graph a line, plot the y-intercept, then use the slope to find another point.

• Positive slope: line rises from left to right.

• Negative slope: line falls from left to right.

Example: The graph of $y = 3x + 1$ passes through (0,1) and rises steeply.

Transformations of Linear Functions

Types of Transformations

• Vertical Shifts: Changing the y-intercept (b) moves the line up or down.

• Changes in Slope: Altering the slope (m) changes the steepness and direction.

Example: $y = x$ vs. $y = x + 2$ (the second is shifted up by 2 units).

Modeling Real-World Situations with Linear Equations

Distance Problems

• Sonya's Distance from Home: Sonya starts 10 miles from home and walks away at 2 miles per hour.

• Equation: $d = 10 + 2t$ where d is distance from home and t is time in hours.

• Graph: The line starts at (0,10) and rises with slope 2.

• Boat Approaching Marina: Boat starts 100 miles away, moves toward marina at 10 miles per hour.

• Equation: $d = 100 - 10t$ where d is distance from marina and t is time in hours.

• Graph: The line starts at (0,100) and falls with slope -10.

Matching Equations to Graphs

Given Functions and Their Graphs

• A. $f(x) = 3x + 1$

• B. $f(x) = x$

• C. $f(x) = -\frac{1}{3}x$

• D. $f(x) = -3x - 4$

• E. $f(x) = -x - 4$

• F. $f(x) = \frac{1}{3}x + 2$

Each function can be matched to its graph by identifying the slope and y-intercept.

Systems of Linear Equations

Definition

A system of linear equations consists of two or more linear equations with the same variables. The solution is the point(s) where the graphs intersect.

Solving by Substitution or Elimination

• Substitution: Solve one equation for one variable, substitute into the other.

• Elimination: Add or subtract equations to eliminate a variable.

Example:

Find the intersection of $f(x) = 5x - 1$ and $g(x) = 6 - x$:

• Set $5x - 1 = 6 - x$

• Solve for x: $6x = 7 \Rightarrow x = \frac{7}{6}$

• Substitute back: $f(\frac{7}{6}) = 5 \times \frac{7}{6} - 1 = \frac{35}{6} - 1 = \frac{29}{6}$

Systems in Standard Form

• Equations can be written as $Ax + By = C$

• Example system:

  • $7x + 2y = -1$

  • $21x + 6y = -3$

• Another system:

  • $-4x + y = 5$

  • $8x - 2y = 3$

Comparison of Linear Equations

Table: Slope and Y-Intercepts of Given Functions

Function	Slope (m)	Y-intercept (b)	Direction
$f(x) = 3x + 1$	3	1	Rises steeply
$f(x) = x$	1	0	Rises moderately
$f(x) = -\frac{1}{3}x$	-1/3	0	Falls gently
$f(x) = -3x - 4$	-3	-4	Falls steeply
$f(x) = -x - 4$	-1	-4	Falls moderately
$f(x) = \frac{1}{3}x + 2$	1/3	2	Rises gently

Summary

• Linear functions are described by $y = mx + b$.

• Slope and y-intercept determine the line's direction and position.

• Real-world problems can be modeled with linear equations.

• Systems of equations are solved by finding intersection points.

• Comparing slopes and intercepts helps match equations to graphs.