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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:Don't forget about the age old question of 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

If you want to learn more check out What is a tradition bearer?

Domain and Range

We also discuss several other topics like What are the three branches of broadway after it has run in nyc?

If you want to learn more check out Characterize the four features of communication.

        Range

(y - values)

Don't forget about the age old question of What makes wound healing is a reparative process, not a regenerative process?

Don't forget about the age old question of Explain how the sympathetic system responds to change.

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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