Stretches and compressions of the parent function
f (
x ) =
bx
For any factor
the function
is stretched vertically by a factor of
if
is compressed vertically by a factor of
if
has a
y -intercept of
has a horizontal asymptote at
a range of
and a domain of
which are unchanged from the parent function.
Graphing the stretch of an exponential function
Sketch a graph of
State the domain, range, and asymptote.
Before graphing, identify the behavior and key points on the graph.
Since
is between zero and one, the left tail of the graph will increase without bound as
decreases, and the right tail will approach the
x -axis as
increases.
Since
the graph of
will be stretched by a factor of
In addition to shifting, compressing, and stretching a graph, we can also reflect it about the
x -axis or the
y -axis. When we multiply the parent function
by
we get a reflection about the
x -axis. When we multiply the input by
we get a
reflection about the
y -axis. For example, if we begin by graphing the parent function
we can then graph the two reflections alongside it. The reflection about the
x -axis,
is shown on the left side of
[link] , and the reflection about the
y -axis
is shown on the right side of
[link] .
Reflections of the parent function
f (
x ) =
bx
The function
reflects the parent function
about the
x -axis.
has a
y -intercept of
has a range of
has a horizontal asymptote at
and domain of
which are unchanged from the parent function.
The function
reflects the parent function
about the
y -axis.
has a
y -intercept of
a horizontal asymptote at
a range of
and a domain of
which are unchanged from the parent function.
Writing and graphing the reflection of an exponential function
Find and graph the equation for a function,
that reflects
about the
x -axis. State its domain, range, and asymptote.
Since we want to reflect the parent function
about the
x- axis, we multiply
by
to get,
Next we create a table of points as in
[link] .
Plot the
y- intercept,
along with two other points. We can use
and
Draw a smooth curve connecting the points:
The domain is
the range is
the horizontal asymptote is
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?