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AU / Mathematics / MATH 1120 / What information can be derived from a graph of a function?

What information can be derived from a graph of a function?

What information can be derived from a graph of a function?

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2.3 - 2.8 STUDY GUIDE:We also discuss several other topics like What do epochal innovations refer to?

09-21-16

2.3 Getting Information from the Graph of a Function:

What can we find?

  • Domain
  • Tange
  • Local Max. and Min.
  • Net Change
  • Increasing and Decreasing Functions
  • Intersections
  • Average rate of change

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We also discuss several other topics like What are the 5 qualities of art and how are they implemented?

Domain and Range

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        Range

(y - values)

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Domain

(x - values)

Ex. 1:

        f(x) =

  • Domain: [-2, 2]
  • Range: [0, 2]
  • Increasing: [-2, 0]
  • Decreasing: [0, 2]

Local Minimums and Maximums

  • The equation has a local maximum at x =2 and x = 6 → (2, 4) and (6, 4) and a local minimum at x = 4 → (4, 0)

Net Change:

What is the net change from x = 2 to x = 3?

        f(6) - f(3) = 4 - 3 = 1

2.4 Average Rate of Change:

Formula:

Examples:

  • f(x) = ; x = 1 and x = 3

  • f(x) = 6/x ; a = a and b = a + h

  • h(t) = t2 + 5t ; t = 1 and t = 3

2.4 Transformations of Functions:

  • Vertical Shifts
  • f(x) + C → shifts up
  • f(x) - C → shifts down
  • Horizontal Shifts
  • g(x- c) → shift to the right
  • g(x+ c) → shift to the left
  • Vertical Stretching:
  • cf(x) → stretch vertically by C
  • -cf9x) → flip upside down and stretch by a factor of the x-axis.
  • Horizontal Stretching
  • f(cx) → stretch horizontally by a factor of 1/c
  • f(-cx) → flip about the y- axis and stretch horizontally by a factor of 1/c.
  • Combining Transformations:
  • f(x) = x3
  • Plot f(-2x) = (-2x)3 = -8x3
  • f(x+1) - 2 ← vertical shift down by -w
  • Horizontal shift left by 1.
  • Even and Odd Functions
  • Even Function:
  • f(x) = f(-x)
  • f(x) = x2
  • f(-x) = (-x)2 = x2
  • Symmetric across y- axis

  • f(x) = x2 + 2
  • f(-x) = (-x)2 + 2 = x2 + 2
  • Therefore, even function

  • Odd Function:
  • f(-x) = -f(x)
  • f(x) = x3
  • f(-x) = (-x)3 = -x3
  • -f(x) = -x3.

  • f(x) = x3 + x
  • f(-x) = (-x)3 + (-x) = -x3 - x
  • -f(x) = -(x3 + x) = -x3 + x
  • Therefore, odd function

2.7 Combining Functions

Sets and Set Notation:

  • Def: A set is a collection of objects, no repeats and no order.
  • Intervals are sets
  • ℝ = (-∞, ∞)
  • ℕ = {1, 2, 3, …..} = {positive whole numbers} = natural numbers
  • A ∩ B = set of all elements in both A and B.
  • Ex. [0, 2] ∩ [1, 3] = [1, 2]
  • If A is a set and is x is some object
  • x ∈ A        x is in A
  • x ∉ A         x is not in A.

  • Ex 1: {x ∈ ℝ | x > 0}
  • = (0, ∞)
  • {x ∈ ℤ | x > 0}
  • ℕ = {1, 2, 3, …..}

  • Basic Operations
  • f and g are functions.
  • Dom(f) = A, Dom(g) = B
  • (f+ g)(x) = f(x) + g(x) → Dom(f+ g) = A ∩ B
  • (f- g)(x) = f(x) - g(x) → Dom(f- g) = A ∩ B
  • (fg)(x) = f(x)g(x) → Dom(fg) = A ∩ B
  • (f/g)(x) = f(x) / g(x) → Dom(f/g) = { x ∈ A ∩ B | g(x) ≠ 0}

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