To simplify, let’s start by factoring out the inside of the function.
By factoring the inside, we can first horizontally stretch by 2, as indicated by the
on the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See
[link] .
Next, we horizontally shift left by 2 units, as indicated by
See
[link] .
Last, we vertically shift down by 3 to complete our sketch, as indicated by the
on the outside of the function. See
[link] .
A function can be shifted vertically by adding a constant to the output. See
[link] and
[link] .
A function can be shifted horizontally by adding a constant to the input. See
[link] ,
[link] , and
[link] .
Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See
[link] .
Vertical and horizontal shifts are often combined. See
[link] and
[link] .
A vertical reflection reflects a graph about the
axis. A graph can be reflected vertically by multiplying the output by –1.
A horizontal reflection reflects a graph about the
axis. A graph can be reflected horizontally by multiplying the input by –1.
A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See
[link] .
A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See
[link] .
A function presented as an equation can be reflected by applying transformations one at a time. See
[link] .
Even functions are symmetric about the
axis, whereas odd functions are symmetric about the origin.
Even functions satisfy the condition
Odd functions satisfy the condition
A function can be odd, even, or neither. See
[link] .
A function can be compressed or stretched vertically by multiplying the output by a constant. See
[link] ,
[link] , and
[link] .
A function can be compressed or stretched horizontally by multiplying the input by a constant. See
[link] ,
[link] , and
[link] .
The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See
[link] and
[link] .
Section exercises
Verbal
When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?
A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.
for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
like this: (2)/(2-x)
the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
functions can be understood without a lot of difficulty.
Observe the following:
f(2) 2x - x
2(2)-2= 2
now observe this:
(2,f(2)) ( 2, -2)
2(-x)+2 = -2
-4+2=-2
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
100•3=300
300=50•2^x
6=2^x
x=log_2(6)
=2.5849625
so, 300=50•2^2.5849625
and, so,
the # of bacteria will double every (100•2.5849625) =
258.49625 minutes
Thomas
158.5
This number can be developed by using algebra and logarithms.
Begin by moving log(2) to the right hand side of the equation like this:
t/100 log(2)= log(3)
step 1: divide each side by log(2)
t/100=1.58496250072
step 2: multiply each side by 100 to isolate t.
t=158.49
Dan
what is the importance knowing the graph of circular functions?
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x:
12x = 144
notice how 12 is being multiplied by x. Therefore division is needed to isolate x
and whatever we do to one side of the equation we must do to the other.
That develops this:
x= 144/12
divide 144 by 12 to get x.
addition:
12+x= 14
subtract 12 by each side. x =2
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.