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3.7 – Linear and Nonlinear Functions

Objectives

  • Identify intervals where a function is increasing, decreasing, or constant.
  • Visually discriminate between graphs of linear and nonlinear functions.
  • Use the horizontal line test to determine whether a function is one-to-one.
  • Explore the graphs of special functions.

 

Key Terms

  • Horizontal – Straight from left to right.
    • On a coordinate plane, the equation of a horizontal line is y = a, where a is a constant.
  • Horizontal Line Test – A method for testing whether the inverse of a given graph is a function.
    • If any horizontal line can be drawn that intersects the graph at two or more points, then the inverse of the graph is not a function.
    • If no such line can be drawn, then the inverse of the graph is a function.
    • A method for testing whether a given graph is a one-to-one function.
  • Many-to-One Function – A function in which 2 or more x-values map to a given y-value.
    • Each x-value has more than one corresponding y-value.
  • One-to-One Function – A function in which each y-value has a single x-value mapped to it.
    • No two x-values map to the same y-value.

 

Notes

  • Linear and Nonlinear Functions
    • A linear function is a graph that demonstrates constant rate of growth or decline
    • A curve (nonlinear function) demonstrates growth or decline that is not at a constant rate

Alg1A 3.07 - Linear Nonlinear04

Alg1A 3.07 - Linear Nonlinear02

 

  • Linear functions are graphs of straight lines, but they must NOT be vertical lines.
    • Vertical lines would fail the vertical line test, obviously.
  • Nonlinear functions are graphs of anything other than straight lines.
    • The graph of a curve that passes the vertical line test is a nonlinear function.
    • The graph of a combination of line segments that passes the vertical line test is a nonlinear function.

Alg1A 3.07 - Nonlinear2

 Nonlinear Function (curved)


  • Review:  What is a function?
    • Function – A graph that passes the vertical line test (each x-value goes to exactly ONE y-value)
      • Draw vertical lines straight up and down on a graph.
        • Do any of the lines cross the graph more than once?
          • If so, it’s not a function.
          • If they ONLY cross once, it IS a function!
    • Relation – Any graph (functions and non-functions)
      • The following graph is a relation, but not a function (it fails the vertical line test in many places)
      • Alg1A 3.07 - Relation
    • One-to-One Function – Passes the vertical line test AND the horizontal line test
      • One input has exactly one output.
      • Alg1A 3.07 - One2One
    • Many-to-One Function – Passes the vertical line test but FAILS the horizontal line test
      • On a graph: If a horizontal line crosses the graph at more than one point.
      • On an input-output table: If any y-value has more than one x-value listed with it.
      • Many inputs can have the same output.
      • Alg1A 3.07 - Many2One

 


  • Constant, Increasing, or Decreasing
    • A linear function has a constant rate of change; however, it may increase, decrease, or be constant over time.
    • To determine if a graph is increasing, decreasing, or constant, look at the graph from left to right.  Does it go up, down, or stay the same?

Alg1A 3.07 - Linear Ex

 

Alg1A 3.07 - Increasing

Increasing

Alg1A 3.07 - Decreasing

Decreasing

 

Alg1A 3.07 - Decreasing2

Decreasing (starts by going down from the left to right; and, even though it goes up and down infinitely, it ends going down on the right)

 

Alg1A 3.07 - Constant Rate

Constant

 


  • Example: Investing
    • An investment is an amount of money you start with to (hopefully) make more money.
    • Interest is a fee charged by a lender or borrower for the use of borrowed money.
    • When you put money in a bank account, the bank pays you interest because it is borrowing your money
      • Simple interest is paid only on the original investment.
        • Each year, Peter’s account would earn 8% of $1000.
      • Compound interest is paid on the original investment and on any interest already earned.
        • Each year, Peter’s account would earn 8% of ($1000 + interest from all previous years).
    • Which is better for Peter, simple or compound interest?

Alg1A 3.07 - Linear Nonlinear03

Answer:  Compound interest is the best option for Peter, because he will earn more money over time.

 

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