f(x) = aˣ a is any value greater than 0 a has a power that contains a variable goes through the point (0, 1) intersecting y axis at 1 never touches the x axis, always greater than 0 Domain: Real Numbers Range: Positive Real Numbers: (0, +∞) aˣ is the inverse function of logₐ(x) (the Logarithmic Function) can be "reversed" by the Logarithmic Function.

Chapter 5: Functions & Graphs Flashcards by Alexandra Miranda

Vertical transformations

stretching and shrinking original function vertically

vertical transformations change only the y coordinate, leaving x alone
multiply the whole function by a number
constant added > 0 expands function
constant added < 0 shrinks function

g(x) = 0.35(x2)

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Horizontal transformations

reflection over they y axis

changes only the x coordinate, leaving y coordinate unchanged
We can flip it upside down by multiplying the whole function by −1
or by multiplying the x in y = 2ˣ by -1
i.e. points (1, 2) become (-1, 2)

g(x) = −(x2)

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Horizontal transformations

shrinking and expanding a function

changes only the x coordinate, leaving y coordinate unchanged
can shrink or expand graph by multiplying x by a number
multiplying by a # > 1 shrinks the function
multiplying by a # < 1 expands the function
whatever number you multiply by, its reciprocal denotes the distance changed

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Vertical transformations

moving graphs up or down

vertical transformations change only the y coordinate, leaving x alone
add a number to or subtract a number from the entire function
constant added > 0 moves function up
constant added < 0 moves function down

original function: (x) = x2

g(x) = x2 + C

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Horizontal transformations

moving graphs left to right

changes only the x coordinate, leaving y coordinate unchanged
can move graph left or right by adding a constant to the x-value
constant added > 0 moves function to the left (the negative direction)
constant added < 0 moves function to the right (the positive direction)

g(x) = (x+C)2

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inverse functions

The inverse of f(x) is f⁻¹(y): used y instead of x to show a different value
shown by putting a “-1” superscript after the function name
read as “f inverse of y”
The inverse of f(x) is f⁻¹(y)
applying a function f and then its inverse f-1 gives us the original value back again
f⁻¹( f(x) ) = x and f( f⁻¹(x) ) = x
when graphed, each is the mirror image of the other, reflected over the line y = x
if a point is (2,4) on one function, the other will have point (4,2)
the domain of one is the range of the other
the range of one is the domain of the other
To be able to have an inverse we need unique values.

Example:

the inverse of f(x) = 2x+3 is: f⁻¹(y) = (y-3)/2
f(4) = 2×4+3 = 11
f⁻¹(11) = (11-3)/2 = 4
“f inverse of f of 4 equals 4”

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logarithmic functions

f(x) = logₐ(x)

When a between 0 and 1:

As x nears 0, it heads to positive infinity
As x increases it heads to negative infinity
f(x) = log_½(x)

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logarithmic functions

f(x) = logₐ(x)

When a > 1:

As x nears 0, it heads to negative infinity
As x increases it heads to positive infinity
f(x) = log₂(x)

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logarithmic functions

f(x) = logₐ(x)

exponential function reversed

a is any value greater than 0, except 1
when a = 1 the graph is not defined
goes through the point (1, 0) intersecting x axis at 1
never touches the y axis, always greater than 0
Domain: Real Numbers
Range: Real Numbers
aˣ (exponential function) is the inverse function of logₐ(x)
can be “reversed” by the Exponential Function

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exponential function

f(x) = aˣ

When a between 0 and 1:

exponential decay function
like exponential growth but in reverse
goes up as x decreases (to left)
goes down as x increases (to right)
models things that shrink over time, like radioactive decay

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exponential function

f(x) = aˣ

When a > 1:

exponential growth
goes higher w/out limit as x increases (to right)
goes lower w/out limit as x decreases (to left)
for figuring out things like investments, inflation, and growing populations

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exponential function

f(x) = aˣ

When a = 1:

graph is a horizontal line @ y = 1

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exponential function

f(x) = aˣ
a is any value greater than 0
a has a power that contains a variable
goes through the point (0, 1) intersecting y axis at 1
never touches the x axis, always greater than 0
Domain: Real Numbers
Range: Positive Real Numbers: (0, +∞)
aˣ is the inverse function of logₐ(x) (the Logarithmic Function)
can be “reversed” by the Logarithmic Function.

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exponential function

when an exponent is an even integer

function values are positive
graph passes through the origin
opening upward

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exponential function

when an exponent is an odd integer

function values are positive when b is positive
function values are negative when b is negative
graph passes through the origin
opening upward

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The only function that is even and odd is

f(x) = 0

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is f(x) = x/(x2−1) Even or Odd or neither?

substitute −x for x:

f(−x) =(−x)/((−x)2−1)

=−x/(x2−1)

=−f(x)

So f(−x) = −f(x) , which makes it an Odd Function

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sine function

odd function
f(x) = sin(x)

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odd function

−f(x) = f(−x) for all x
has origin symmetry
called “odd” because the functions x, x³, x⁵, x⁷, etc behave like that
sine functions are also odd: sin(x)
odd exponents don’t always make an odd function: x³+1 is not an odd function
cannot contain a constant term: y = x³ - 5x + 2

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cosine function

even function
like Sine, but starts at 1, heads down until π radians (180°) and then up again

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absolute value function

g(x) = |x|
even function
makes a right angle at (0,0)
Domain: Real Numbers
Range: Non-Negative Real Numbers: [0, +∞)
can be a piecewise function
Are you absolutely positive? Yes! Except when I am zero

parabolic function

f(x) = x²
f(x) = x²+1
even function

even functions

f(x) = f(−x) for all x
there is symmetry about the y-axis (like a reflection)
called “even” functions because the functions x2, x4, x6, x8, etc behave like that: y = 9x⁴ - 4x² + 3
But an even exponent does not always make an even function: example (x+1)² is not an even function
constant term like 3 is the same as 3x⁰ and zero is even
y = cos(x) is also an even function

to use point-slope form you need to know 2 things:

a point on a line
the line’s slope
y - y₁ = m(x - x₁)
y - 11 = 3(x - 2) → y = 3x + 5

point-slope form

identity function

* if m = 1 and b = 0: y = 1x + 0 → y = x * lines go through the origin (0, 0) * makes a 45 * ° * angle * outputs are the same as the inputs

constant function

* horizontal line * has an equation of y = 10 * has a slope of 0

equation of a horizontal line technically fits the form y = mx + b. explain.

* slope (m) of a horizontal line is 0 * 0 times x = 0 so y = b * y = 10 → y = 0x + 10

equation of a horizontal line

y = 10

all lines except for _____ lines can be written in slope-intercept form, they are written like _____ where the number tells you where the vertical line crosses the \_\_\_-\_\_\_\_\_

* vertical * x = 6 * x-axis

slope intercept form

In general, the slope intercept form assumes the formula: y = mx + b. * m is the slope * b is the y -intercept

y-intercept

if slope of a line is 3, the perpendicular line has a slope of

1/3

parallel lines have the ____ slope. perpendicular lines have _____ \_\_\_\_\_ slopes.

* same * opposite perpendicular

formula for slope

lines that go up to the right have a _____ slope. lines that go down to the right, have a _____ slope. horizontal lines have a slope of \_\_\_\_\_, and vertical lines _____ \_\_\_\_\_\_ a slope, it is \_\_\_\_\_

* positive * negative * zero * don't have * undefined

How does a linear function graph look like

tangent and normal curve

tangent

a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at another point

relation

any collection of points on the x, y coordinate system

vertical line test

* a curve is a function if a vertical line drawn through the curve, regardless of where it's drawn, touches the curve only once * this guarantees that each input w/in the function's domain has exactly one output

composite function

* combination of 2 functions * always calculate the inside function first * f(x) = x², g(x) = 5x - 8 * (f ⚬ g) = f(g(3)) = 49

function notation

* replace "y" with "f(x)": f(x) = 5x³ - 2x² + 3 * f(x) reads as "f of x"

independent variable

* a variable whose variation does not depend on that of another * listed on x axis

because you plug numbers into the independent variable it's also called the _____ \_\_\_\_\_\_

input variable

dependent variable

* the thing that depends on the other thing, the independent variable * also called the output variable * y variable, on y axis what we're usually more interested in

range

set of all outputs of the function

domain

set of all inputs of a function

a function has ______ \_\_\_\_\_ output for each input

only one

basically a function is

a relationship between 2 things in which the numerical value of one thing in some way depends on the value of the other

Differential calculus involves finding the _____ or _____ of various functions and integral function involves computing the _____ \_\_\_\_\_\_ functions.

* slope * steepness * area underneath