Recognizing function families

About this lesson

Integrated Math III › Functions and modeling

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Recognizing function families

Warm-up

Three quick questions from earlier lessons.

Discover

f(x) = x² and g(x) = 2^x: which one has the variable in the exponent?

  1. g
  2. f

Key idea

Recognize the family from the formula, the table or the graph.

Polynomial: sums of terms ax^n (x in the base, whole-number powers). Rational: a quotient of polynomials (x in a denominator; vertical asymptotes). Exponential: x in the exponent (equal ratios in a table). Logarithmic: log of x (slow growth, vertical asymptote). Trigonometric: sine or cosine of x (repeats periodically).

Worked example

Name the families of 3x³ − x, 5/(x − 2), 4 · (0.5)^x, ln x and sin 2x.

polynomial, rational, exponential, logarithmic, trigonometric
Formulas.

Your turn, guided

Your turn: a table goes 1, 3, 9, 27. By what factor does it grow each step?

33
3 ÷ 1, 9 ÷ 3.

Practice ★

  1. What kind of function is f(x) = 9 · 3^x?
    1. Exponential
    2. Polynomial
    3. Linear
  2. What kind of function is f(x) = 4 · 5^x?
    1. Exponential
    2. Polynomial
    3. Linear
  1. What kind of function is g(x) = (x + 4) / (x − 2)?
    1. Rational, with a vertical asymptote at x = 2
    2. Polynomial of degree 1
    3. Logarithmic

Watch out!

Practice ★★

  1. A function has f(0) = 3, f(1) = 6, f(2) = 12, f(3) = 24. Which family does it most likely belong to?
    1. Exponential: the values multiply by 2 each step
    2. Quadratic: the values grow faster and faster
    3. Linear
  2. A function has f(0) = 4, f(1) = 8, f(2) = 16, f(3) = 32. Which family does it most likely belong to?
    1. Exponential: the values multiply by 2 each step
    2. Quadratic: the values grow faster and faster
    3. Linear
  1. Boss problem. Two models predict the number of app users (in thousands) after t months: A(t) = 37t + 4 and B(t) = 4 · 2^t.

Exit ticket

Three questions to check you have got it.

  1. What kind of function is f(x) = 9 · 3^x?
    1. Exponential
    2. Polynomial
    3. Linear
  2. What kind of function is f(x) = 4 · 5^x?
    1. Exponential
    2. Polynomial
    3. Linear
  3. What kind of function is f(x) = 9 · 5^x?
    1. Exponential
    2. Polynomial
    3. Linear

Summary card

Function families

Base or exponent, denominator, log, periodic: each family has its sign.

Watch out: x in the exponent means exponential, not polynomial.

End of the lesson

Next lesson: Average rate of change; where a function increases or decreases Back to the chapter

Answers: Recognizing function families

A

  1. Exponential
  2. Exponential

B

  1. Rational, with a vertical asymptote at x = 2

C

  1. Exponential: the values multiply by 2 each step
  2. Exponential: the values multiply by 2 each step

D

  1. 3030

E

  1. Exponential
  2. Exponential
  3. Exponential

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