Lines and Graphing

Graphing Equations of Lines

Graphing linear equations is a fundamental skill in College Algebra. The most efficient method is to rewrite the equation in slope-intercept form.

• Slope-Intercept Form: , where m is the slope and b is the y-intercept.

• Example: Graph .

  • Rewrite:

  • Slope: ; y-intercept:

• Slope Shortcut: For , numerator is vertical movement, denominator is horizontal.

• Intercept Method: Set for x-intercept, for y-intercept.

• Quiz Example: Given , solve for :

  • Slope: ; y-intercept:

Functions and Graph Interpretation

Ordered Pairs & Reading a Function Graph

Understanding how to check solutions and interpret function graphs is essential.

• Checking Solutions: Substitute values into the equation; if both sides match, the point is a solution.

• Example: Does solve ? ; Yes.

• Negative Substitutions: Always use parentheses: .

• Reading from a Graph: Find on the axis, move to the graph, read the y-value.

• Zeros/Solutions: means find x-intercepts.

• Increasing/Decreasing/Constant: Read left to right; rising = increasing, falling = decreasing, flat = constant.

Function Evaluation

Evaluating Functions

To evaluate a function, replace every occurrence of with the given input.

• Main Rule: Replace all with the input value, using parentheses.

• Example 1: , find :

• Example 2:

• Common Mistake: Replace every , not just one.

Domains of Functions

Finding Domains

The domain of a function is the set of all allowable input values (x-values).

• Polynomial: No restriction; domain is all real numbers.

• Fraction: Denominator cannot be zero.

• Square Root: Inside must be .

• Cube Root: No restriction.

• Square Root in Denominator: Inside must be .

• Example (Fraction): ; domain: ;

• Example (Square Root): ; domain: ;

Domains with Multiple Restrictions

Some functions have more than one restriction.

• Example:

  • Square root:

  • Denominator:

  • Combined domain:

• Interval Notation: Use [ or ] for included endpoints, ( or ) for excluded. Infinity always uses parentheses.

Slope and Equations of Lines

Slope Formula and Line Forms

Slope measures the steepness of a line. There are several forms for linear equations.

• Slope Formula:

• Example: Points and :

• Line Forms:

  • Slope-Intercept:

  • Point-Slope:

  • Standard:

• Finding a Line Through Two Points: Use slope and one point in point-slope form, then convert if needed.

Parallel and Perpendicular Lines

Relationships of Slopes

Parallel and perpendicular lines have specific relationships between their slopes.

• Example: Original line ; perpendicular slope .

• Fast Check: If original slope is , perpendicular is .

Rates of Change and Linear Models

Rate of Change

In linear models, the rate of change is the slope, often described in real-world terms.

• Rate of Change:

• Example: 20 customers at month 1, 50 at month 4:

• Interpretation: Company gains 10 customers per month.

• In : is the rate, is the starting value.

Linear Modeling Word Problems

Translate real-world scenarios into equations by defining variables and translating statements.

• Example: "Company A completed two less than two times as many projects as Company B."

  • Substitute:

• Common Words and Math Ideas:

  Word	Math Idea
  sum / total	+
  difference	-
  twice
  5 more than x
  5 less than x
  is	=
  at least
  at most
  more than
  less than

Linear Inequalities

Solving Linear Inequalities

Linear inequalities are solved like equations, but the inequality sign must be flipped when multiplying or dividing by a negative.

• Big Rule: Flip the sign if you multiply/divide both sides by a negative.

• Example:

• Interval Notation:

Piecewise Functions

Understanding Piecewise Functions

Piecewise functions use different formulas for different intervals of x.

• Main Idea: Choose the correct formula based on the value of .

• Example:

  • for

  • for

  • (use )

  • (use )

• Boundary Points: or means open circle; or means closed circle.

• Graphing: Use open/closed circles at endpoints; apply each rule only on its interval.

• Setting Up: For word problems, specify formula and interval for each piece.

Formula Sheet and Test Strategies

Essential Formulas

• Parallel: same slope

• Perpendicular: negative reciprocal

• Divide/multiply inequality by negative: flip sign

• Denominator

• Inside square root

Test Checklist

• Show enough work

• Use parentheses for negative substitutions

• Keep subtraction order in slope formula

• Flip and change sign for perpendicular slopes

• Remove denominator zeros for domains

• Require inside for square roots

• Flip sign for inequalities after dividing/multiplying by negative

• Choose correct rule for piecewise functions

• Report x/y values correctly for graph questions

• Use brackets for included endpoints

Mastery Check: Practice Problems

• Graphing: Convert to slope-intercept form:

• Ordered Pair: Does solve ? Yes.

• Function Evaluation: , find

• Domain: ; domain:

• Square-root Domain: ; domain:

• Slope: Through and :

• Equation of Line: Slope $2(3,4)y = 2x - 2$

• Parallel: Parallel slope to is

• Perpendicular: Perpendicular slope to is

• Inequality: Solve : ; interval

• Piecewise: for , for ; , ,