Pre-Cal Boot Camp: Student PacketParent Functions

Parent Function Graph Parent Function Graph

Lineary=x

Domain: (-∞,∞)Range: (-∞,∞)

Symmetry: Odd Origin

Absolute Valuey=IxI

Domain: (-∞,∞)Range: [0,∞)

Symmetry: Even Y-axis

Quadraticy=x2

Domain: (-∞,∞)Range: [0,∞)

Symmetry: Even Y-axis

Radicaly=√ x

Domain: [0,∞)Range: [0,∞)

Symmetry: Neither

Cubicy=x3

Domain: (-∞,∞)Range: (-∞,∞)

Symmetry: Odd Origin

Cube Rooty=3√ x

Domain: (-∞,∞)Range: (-∞,∞)

Symmetry: Odd Origin

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Exponentialy=bx, b>1

Domain: (-∞,∞)Range: (0,∞)

Symmetry: Neither

Logy=log b x , b>1, x>0

Domain: (0,∞)Range: (-∞,∞)

Symmetry: Neither

Rational(Inverse)y=1/x

Domain: (-∞,0)U(0,∞)

Range: (-∞,0)U(0,∞)

Symmetry: Odd Origin

Rational(InverseSquared)

y=1/x2

Domain: (-∞,0)U(0,∞)

Range: (0,∞)

Symmetry: Even Y-axis

Greatest Integery=int(x)=[x]

Domain: (-∞,∞)Range:{y:yεZ}

(integers)

Symmetry: Neither

Constanty=C (in this graph

y=2)

Domain: (-∞,∞)Range: {y: y=C}

Symmetry: Even Y-axis

Tips and Tricks - Transformations of Functions

y=mx+v

y=a ( x−h )2+v

y=a√x−h+v

y=a ( x−h )3+v

y=a 3√x−h+v

y=a|x−h|+v

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y=aⅇ( x−h)+v

y=a 2( x−h )+v

y=a log10 ( x−h )+v

y= ax−h

+v

y= a( x−h)2 +v

Tips and Tricks – Vertical Asymptotes and Horizontal Asymptotes

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f ( x )=1x

f ( x )= 1x−1 f ( x )= x2

x−1

Tips and Tricks – For x and y Intercepts, Vertical Asymptotes, and Horizontal Asymptotes

1. y=x+1

2. y=x2+3

3. y=x3+6

4. y=√x

5. y=√x−9

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6. y= 3√x

7. y= 3√x+10

8. y=1x

9. y= 1x−1

10. y= 1x2

11. y= 1( x+4 )2

12. y= 1√ x

13. y= 1√ x−8

Tips and Tricks – End Behavior and Multiplicity

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Quadraticy=x2

Domain: (-∞,∞)

Range: [0,∞)

Symmetry: Even Y-axis

Quadraticy=−x2

Domain: (-∞,∞)

Range: [0,-∞)

Symmetry: Even Y-axis

Cubicy=x3

Domain: (-∞,∞)

Range: (-∞,∞)

Symmetry: Odd Origin

Cubicy=−x3

Domain: (-∞,∞)

Range: (-∞,∞)

Symmetry: Odd Origin

Multiplicity (repeated zeros):

A factor of (x - a)k, k > 1, yields a repeated zero x = a of multiplicity k.

1. If k is odd, the graph crosses the x-axis at x = a.2. If k is even, the graph touches the x-axis at x = a.

1. y=x

2. y=x2

3. y=x ( x−1 )

4. y=x2 ( x−3 )4

5. y=x3 ( x−8 )2

6. y=−x3 ( x−6 )

7. y=−x2 ( x+7 )

8. y=x2−81

9. y=x2−6x+9

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10. y=x2+2 x−6

11. y=x3−25 x

12. y=x3−x2+2 x−2

13. y=x3+27

Tips and Tricks – Inverse Functions and the Horizontal Line TestFinding the Inverse of a Function:

The equation for the inverse of a function f can be found as follows:

1. Replace f(x) with y in the equation for f(x)2. Interchange x and y3. Solve for y. If this equation does not define y as a function of x, the function f does not have an

inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.

4. If f has an inverse function, replace y in step 3 by f-1(x). We can verify our result by showing that f(f-1(x))=x

REMEMBER: The Horizontal Line Test and One-to-One Functions:

A function f has an inverse that is a function, f-1, if there is no horizontal line that intersects the graph of the function f at more than one point.

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1. f ( x )=7 x−52. f ( x )=4 x3−13. f ( x )=x2

4. f ( x )=5x+4

Tips and Tricks – Odd and Even FunctionsDefinition of Even and Odd Functions:

The function f is an even function if

f(-x)=f(x) for all x in the domain of f.

The right side of the equation of an even function does not change if x is replaced with -x.

The function f is an odd function if

f(-x)=-f(x) for all x in the domain of f.

Every term in the right side of the equation of an odd function changes its sign if x is replaced with -x.

1. f ( x )=x3− x2. f ( x )=x4−x2

3. f ( x )=x2+7

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4. f ( x )=x3+4

Tips and Tricks – Determining the Following for GraphsUse the graph of f to determine each of the following. Where applicable, use interval notation:

a. The domain of fb. The range of fc. The x-interceptsd. The y-intercepte. Intervals on which f is increasingf. Intervals on which f is decreasingg. Intervals on which f is constanth. The number at which f has a relative minimumi. The relative minimum of fj. f(-3)k. the values of x for which f(x)=-2l. Is f even, odd, or neither?

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Tips and Tricks – Difference QuotientDefinition of a Difference Quotient:

The expression

f ( x+h )− f ( x )h

For h ≠ 0 is called the difference quotient.

1. For f ( x )=4 x , find f ( x+h )−f ( x )

h

2. For f ( x )=x2−4 x+7 , find f ( x+h )−f ( x )

h

3. For f ( x )=1x−2 , find

f ( x+h )−f ( x )h

4. For f (x)=√x +8, find f ( x+h )−f ( x )

h

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Tips and Tricks -For Deriving Pythagorean Identities

sin2 (θ )+cos2 (θ )=1

tan2 (θ )=sec 2 (θ )−1

cot2 (θ )=csc2 (θ )−1

Tips and Tricks – For Even and Odd Trig Identitiescos (−θ )=cos (θ )

sec (−θ )=sec (θ )

sin (−θ )=−sin (θ )

csc (−θ )=−csc (θ )

tan (−θ )=−tan (θ )

cot (−θ )=−cot (θ )

Trigonometric Parent Functions

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Blank Unit Circle

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Tips and Tricks – Do’s and Don’ts of Calculator:

The Do’s:

Texas Instruments TI - 84 Plus CE: Casio FX-115 ES Plus:

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Best Software Calculator!!!TI SmartView CE for the TI – 84 Plus Family:

The Don’ts:Texas Instruments TI-83 series calculators

Texas Instruments TI-30 series calculators

Texas Instruments TI-nSpire calculators

Casio FX-9750 GII calculator

Casio G series calculators

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Formula Trig Sheet

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