Graphing Summer School

Functions and Their Properties Restricted Domains 1. y =- 4x - 3 2. y = √(4x - 3) 3. y = 1 4x - 3 4. A(s) = √(3) s2 4 where A(s) is the area of an equilateral triangle with sides of length s. 5. y = log (2x) 6. y = 4x

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Finding Range y= 4 x-1

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Continuity: continuous at all x

removable discontinuity

jump discontinuity

infinite discontinuity

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A function f is continuous at x = a if lim f(x) = f(a). A function f is discontinuous at x = a if it is not continuous at x = a.

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Increasing and Decreasing Functions Increasing

Decreasing

Constant

Mixed

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Increasing on an interval if, for any two points in the interval, a positive change in x results in a positive change in y. Decreasing on an interval if, for any two points in the interval, a positive change in x results in a negative change in y. Constant on an interval, if for any two points in the interval, a positive change in x results in a zero change in y.

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Boundedness Bounded below - if there is a number b less than or equal to every number in the range of f. That number b is called a lower bound of f. Bounded above - if there is a number B greater than or equal to every number in the range of f. That number B is called an upper bound of f.

f is bounded if it is bounded both above and below.

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Look at these examples: y = 3x2 - 4 y= x 1 + x2

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Local and Absolute Extrema Local Maximum f(c) that is greater than or equal to all of the range values of f on some open interval containing c. If f(c) > all of the range values of f, then f(c) is a maximum (or absolute maximum) value of f.

Local Minimum f(c) < all of the range values of f on some open interval containing c. If f(c) < all of the range values of f, the f(c) is a minimum (or absolute minimum) value of f.

Local extrema are also called relative extrema

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Look at the graphs: y = (x - 1)2 + 4 y = x4 - 7x2 + 6

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Symmetry symmetric with respect to the y-axis even functions f(-x) = f(x)

symmetric with respect to the origin odd functions -f(x) = f(-x)

symmetric with respect to the x-axis not a function (x, -y) whenever (x, y)

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Examples 1. y = -x2 + 5 2. y =

x3 4 - x2

3. y = (x - 5)2 + 1 4. y2 = x - 2

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Asymptotes:

y = 2x2 4 - x2 vertical: horizontal:

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horizontal: y = b if f(x) → b as x → + ∞ lim f(x) = b lim f(x) = b x→∞ -

y=

x→∞+

x x2 - x - 2

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vertical: x = a if f(x) → + ∞ as x → a lim f(x) = + ∞ lim f(x) = + ∞ x→a-

x→a+

End Behavior: 1. y = 3x x2 + 1 2. y = 3x2 x2 + 1 3. y = 3x3 x2 + 1 4. y = 3x4 x2 + 1

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12 Basic Functions f(x) = x

f(x) = x2

f(x) = x3

f(x) = 1 x

f(x) = x

f(x) = ex

f(x) = ln x

f(x) = sin x

f(x) = cos x

f(x) = |x|

f(x) = int (x)

f(x) =

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1 1 + e-x

Look at the 1. domain and range 2. continuity 3. boundedness 4. symmetry

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Practice domain and range: 1. f(x) = x2 - 5

2. f(x) = |x - 4|

3. h(x) = ln(x + 6)

4. f(x) = 1 + 3 x

5. m(x) = int (x/2)

6. p(x) = (x + 3)2

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Find a. the interval where it is increasing or decreasing b. even, odd, neither c. extrema 1. f(x) = √(x - 10)

2. f(x) = 4sin x

3. f(x) =

4. q(x) = ex + 2

5. f(x) = |x| - 10

1 + e-x 6. f(x) = 4 cos x

7. m(x) = |x - 2|

8. f(x) = 2 - |x|

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3

Graphs

Translations horizontal y = f(x - c) y = f(x + c) vertical

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y = f(x) + c y = f(x) - c

y = (x + 2)2 - 4

Reflections x-axis y-axis

(x, y) → (x, -y) (x, y) → (-x, y)

Find the equation of the reflection of y = 5x - 9 x2 + 3 across the x-axis and the y-axis.

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Vertical and Horizontal Stretches and Shrinks horizontal stretches and shrinks y = f(cx) if c < 1, stretches horizontal by a factor of 1/c if c > 1, shrinks horizontal by a factor of 1/c

vertical stretches and shrinks y = cf(x) if c > 1, stretches vertical by a factor of c if c < 1, shrinks vertical by a factor of c

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Example: f(x) = x3 - 16x a. find an equation that is a vertical stretch of f(x) by a factor of 3

b. find an equation that is a horizontal shrink of f(x) by a factor of ½

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Combine the Transformations y = x2 horizontal shift 2 to the right vertical stretch by a factor of 3 vertical translation 5 units up

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What transformations are taking place and in what order? 1. y = 2 + 3 f(x + 1) 2. y = -f(x + 1) + 1

3. y = f(2x)

5. y = 3f(2x - 1) + 4

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4. y = 2f(x - 1) + 2

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