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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? Description

##### Description: A comprehensive study guide of 2.3-2.7 to help you prepare for the test on Monday, including examples from SI sessions and the practice test. I will upload the notes of 2.8 tomorrow when we have finished going over that section. Good luck!
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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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