Aim: What concepts have we available to aide us in sketching functions?

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Aim What concepts have we available to aide us
in sketching functions?
Do Now
Find the domain of
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Concepts used in Sketching

• x- and y-intercepts

• symmetry

• domain range

• continuity

• vertical asymptotes

• differentiability

• relative extrema

• concavity

• points of inflection

• horizontal asymptotes

Use them all? If not all, which are best?
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Guidelines for Analyzing Graph
1. Determine the domain and range of the
function.
2. Determine the intercepts and asymptotes of
the graph.
3. Locate the x-values for which f(x) and
f(x) are either zero or undefined. Use the
results to determine relative extrema and points
of inflection.
Also helpful symmetry end behavior
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Abridged Guidelines the 4 Tees
T1 Test the function
T2 Test the 1st Derivative
T3 Test the 2nd Derivative
T4 Test End Behavior
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Model Problem 1
Analyze the graph of
1. find domain range
exclusions at zeros of denominator
domain all reals except 2
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Model Problem 1
Analyze the graph of
2. find intercepts asymptotes
y-intercept
x-intercept
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Model Problem 1
Analyze the graph of
2. find intercepts asymptotes
verticals asymptotes found at zeros of denominator
x 2
horizontal asymptote
If degree of p degree of q, then the line y
an/bm is a horizontal asymptote.
y 2
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Model Problem 1
Analyze the graph of
3. find f(x) 0 and f(x) 0 or undefined
x 0
(x2 4)2 0
undefined at zeros of denominator
x 2
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Model Problem 1
Analyze the graph of
3. find f(x) 0 and f(x) 0 or undefined
no real solution
no possible points of inflection
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Model Problem 1
3. test intervals
f(x) f(x) f(x) characteristic of Graph
-? lt x lt -2
x -2 Undef Undef Undef
-2 lt x lt 0
x 0 9/2
0 lt x lt 2
x 2 Undef Undef Undef
2 lt x lt ?
decreasing, concave down
decreasing, concave up


relative minimum
0
increasing, concave up


increasing, concave down

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Model Problem 1
(0, 9/2) relative minimum
increasing, concave down 2 lt x lt ?
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Model Problem 2 What the cusp!!
Analyze the graph of
T1
Find Domain
all reals
Find intercepts asymptotes
no vertical or horizontal asymptotes
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Model Problem 2 What the cusp!!
Analyze the graph of
T2
1st Derivative Test
x at 0 is undefined
BUT . . .
f gt 0 inc
f lt 0 dec
x 0 is defined for original function
a cusp!!!
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Model Problem 2 What the cusp!!
Analyze the graph of
T3
2nd Derivative Test
x at 0 is undefined
f gt 0 con up
f gt 0 con up
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Model Problem 3
lt 0 dec
lt 0 dec
gt 0 inc
gt 0 inc
gt 0 c.u.
lt 0 c.d.
gt 0 c.u.
lt 0 c.d.
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Model Problem 4
Analyze the graph of
1. find Domain
2. find intercepts asymptotes
verticals asymptotes found a zeros of denominator
x 2
1 sin x 0 sin x -1