Power Function:
Exercise: Recognize and analyze the graph of the power function as the exponent "n" and the coefficient "a" vary
1. Consider a positive exponent power function
A) Describe the graph of the function when n is even, for this you must attach an image of the graph of the function that meets these conditions.
B) Describe the graph of the function when n is odd, for this you must attach an image of the graph of the function that meets these conditions.
C) Describe what happened when the coefficient a is positive and also when it is negative. Is there any change in the graph?
D) Describe what happened when the coefficient a increases or decreases. What happens to the graph of the function?
2. Consider a negative exponent power function
A) Describe the graph of the function when n is even, for this you must attach an image of the graph of the function that meets these conditions.
B) Describe the graph of the function when n is odd, for this you must attach an image of the graph of the function that meets these conditions.
C) Describe what happened when the coefficient a is positive and also when it is negative. Is there any change in the graph?
D) Describe what happened when the coefficient a increases or decreases. What happens to the graph of the function?
Answers
Problem 1
Part A
If n is even, then we have a "parabolic" shape going on. I put that in quotes because if n = 2, then we have a true parabola; however, if n is even and larger than 2, then we have a curve that looks like a parabola, but it's not exactly a perfect parabola. All of these curves have in common that the end behavior is the same for the left and right sides.
Consider the example where a = 1 and n = 2. This means
y = a*x^n
turns into
y = 1*x^2
which is the same as
y = x^2
This produces the parent parabola curve. This curve opens upward.
I recommend using Desmos to graph y = a*x^n, where the 'a' and n are parameters set up. Check out the screenshot below.
---------------------
Part B
If n is odd, then we'll get something that looks like a cubic curve. If n = 3, then we get a cubic curve. If n is odd and n > 3, then we get something that looks like a cubic curve, but it's not exactly this perfect cubic.
If we let a = 2 and n = 3, then y = a*x^n becomes y = 2x^3. The graph of which is below.
---------------------
Part C
If a > 0 then the curve is like the previous examples shown earlier.
If a < 0, then we will reflect those given earlier curves over the horizontal x axis. An example is shown below using y = -x^2. This time we have a = -1 and n = 2. In other words, I reflected the example in part A over the x axis.
---------------------
Part D
As 'a' increases, then the graph gets more narrow. It also gets vertically stretched. Consider the example of going from y = 1x^2 to y = 3x^2 as shown below.
As 'a' decreases, then the process is done in reverse: the curve gets more flatter and wider horizontally. We consider the curve to be vertically compressed or squished. So we'd go from y = 3x^2 to y = 1x^2 as an example.
============================================================
Problem 2
Part A
For all of problem 2, we'll only consider negative values of n. We want n to be even, so let's say n = -2. Furthermore, let's say a = 2.
This means y = ax^n becomes y = 2x^(-2) which is the same as y = 2/(x^2)
The curve we get is really two separate pieces. There's a gap or disconnect at x = 0 because of a division by zero error. One piece of the curve is always increasing, while the other piece is always decreasing.
---------------------
Part B
Since we want n to be odd and negative, let's say n = -1. We'll keep a = 2 the same as before.
That means y = ax^n becomes y = 2x^(-1) as shown in the diagram below. What we have is a hyperbola. This is a disconnected set of two curves that collectively entirely define this function in a visual sense. Note the gap when x = 0 due to the division by zero error.
Recall that y = 2x^(-1) is the same as y = 2/x
Again, I'm using Desmos to generate each of these graphs.
Similar to part A, the two pieces are disconnected. This curve is known as hyperbola. It's always increasing if a < 0, and it's always decreasing when a > 0.
---------------------
Part C
If 'a' is positive, then the graph of y = 2/(x^2) is entirely above the x axis (aka everything is positive). On the flip side, if 'a' is negative, then everything swaps to being negative. That will reflect the curve over the x axis to get what you see in the example below. We have a = -2 and n = 2.
---------------------
Part D
As 'a' increases, this means the graph spreads out. If 'a' decreases, then it gets more compressed.
An example is shown below going from y = 1/x to y = 3/x which shows expansion going on (reverse it to get a compression).
============================================================
Once again, the screenshots are shown below as examples.
Related Questions
2. Which type of variation is represented by the following equation?
Answers
indirect variation
Verification
[tex]\\ \rm\Rrightarrow s\propto \dfrac{1}{y}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{s_1}{t_2}=\dfrac{s_2}{t_1}[/tex]
[tex]\\ \rm\Rrightarrow s_1t_1=s_2t_2[/tex]
find the value of x
x = [?]
Answers
Answer
4
Step by step explanation
2/3=x/10-x
cross multiply
2(10-x)=3x
20-2x=3x
20=3x+2x
20=5x
x=4
12.5 x 0.8
(without rounding and pls help me)
Answers
So answer
12.5* 0.8 = 10
Answer:
That’s simple 10
Step-by-step explanation
Let me know if I’m correct
hope this helps
Note: Enter your answer and show all the steps you use to solve this problem in the space provided. 6 × 5 + 3 × 8
Make it simple, please
Answers
Answer:
54
Step-by-step explanation:
6x5=30
3x8=24
30+24=54
The ratio of sugar to flour in Sydney's favorite recipe for chocolate chip cookies is 3 to 2. If Sydney used 20 tsp of flour, how many tsp of sugar did she use?
Answers
Answer:
30 tsp of sugar
Step-by-step explanation:
If 20 is 2 parts of the 5 parts then 1 part is 10
10x3 =30
30 to 20 = 3 to2
help plsss . 10 points !
Answers
Answer:
B
Step-by-step explanation:
(7×2)⁶
when we put a number to the power of something, we need to include the whole number. bit just a part of it.
so, it must be 7⁶×2⁶
just think about the simple example 6², which we could write as (2×3)².
would it be sufficient to e.g. square only one of the factors ?
6² = 36
but e.g. 2×3² = 18. so, that's really not it.
or add the two factors and then square them ?
2+3 = 5. 5² = 25. so, that's not it either.
or multiply the exponent in ?
2×3×2 = 12. so that's not it either.
no, it truly is you need to do the operation to all parts.
2²×3² = 4×9 = 36. yes, that fits.
therefore, B is the right answer.
What is the range of exponential function g?
Answers
g(x) > -6
The range of the exponential function is: B. [tex]g(x)>-6[/tex]
Recall:
Range of any function includes all possible values of y (output)
Domain of any function includes all possible values of x (input).
Thus:
The values of y in the exponential function greater than -6 on the y-axis as shown in the graph given.
Therefore:
Range of the exponential function given in the graph is: B. [tex]g(x)>-6[/tex].
Learn more about exponential function here:
https://brainly.com/question/19554225
ANSWER FAST 50 POINTS PLEASE !!!
Identify the property that justifies each step asked about in the answer area below.
Line 1: (9+x)x
Line 2: x(9+x)
Line 3: x(x+9)
Answers
Line 1-Line 2-Associative property .
Line 2-Line 3- Distributive property
Solution:-
[tex]\\ \rm\longmapsto (9+x)x[/tex]
[tex]\\ \rm\longmapsto x(x+9)[/tex]
[tex]\\ \rm\longmapsto x(9+x)[/tex]
[tex]\\ \rm\longmapsto x^2+9[/tex]
what is the solution of linear equation of cosx(e^2y-y)=e^y sin2x
Answers
This is college? Woah I'm not advanced. Middle school!
Calculate 20% of 3 3/4 years in months.
Answers
Answer:
It is 9 months
Step-by-step explanation:
[tex] = 20\% \times 3 \frac{3}{4} \\ \\ = \frac{20}{100} \times \frac{15}{4} \\ \\ = \frac{300}{400} \\ \\ = \frac{3}{4} \: \: { \sf{years}}[/tex]
convert them to months by multiplying by 12:
[tex]{ \sf{ = \frac{3}{4} \times 12}} \\ \\ = { \sf{9 \: months}}[/tex]
create 5 problems using all operations that has a solution of 5
Answers
Answer:
((5*5) + (10-5) * (50/100)) ÷ 3 = 5
((4*4)+(17-8 *(1.378/2)) ÷ 3.445 = 5
Step-by-step explanation:
Consider the set A with n(A) = 20. How many subsets could be formed from this set?
Answers
Answer:
There are [tex]2^{20}[/tex] subsets of [tex]A[/tex]
Step-by-step explanation:
Using the formula for the number of subsets of a given (finite) set, the number of subsets of [tex]A[/tex] is
[tex]2^{n(A)}=2^{20}[/tex]
50 Points to correct answer!!!
what is the average rate of change from 2 to 9 of the function represented by the graph?
Answers
Answer:
-3/7
Step-by-step explanation:
It is asking to find the slope of the secant line going through points (2,f(2)) and (9,f(9)).
We must find f(2) by looking at the curve at x=2. We should see that y=2 there so f(2)=2.
We must find f(9) by looking at the curve at x=9. We should see that y=-1 there so f(9)=-1.
The slope of a line is calculated by finding the ratio of the change of y to the change of x.
(-1-2)/(9-2)
(-3)/(7)
-3/7
Find the midpoint of the line segment with the given endpoints.
(-1, 5), (6,6)
A) (-3.5, -0.5)
C) (13,7)
B) (2.5, 5.5)
D) (2,6)
Answers
Could someone please solve this using a^2+b^2=c^2
Answers
Step-by-step explanation:
it is shown in the above process.
hope you understand
what fraction of a minute is 50 seconds
Answers
[tex]\large \mathfrak{Solution : }[/tex]
we know,
[tex]60 \: \: sec = 1 \: \: min[/tex][tex]1 \: \: sec = \dfrac{1}{60} \: \: min[/tex][tex]50 \: \: sec = \dfrac{1}{60} \times 50 \: \: min[/tex][tex]50 \: \: sec = \dfrac{5}{6} \: \: min[/tex]50 seconds is 5/6 th of a minute.
find the real numbers x&y so that (x^2+2xy)+i(y-1) = (x^2-2x+2y) - i(x+y)
Answers
Answer:
[tex]\displaystyle x_1 = 2-\sqrt{3} \text{ and } y_1 = \frac{\sqrt{3}-1}{2}[/tex]
Or:
[tex]\displaystyle x _ 2 = 2 + \sqrt{3} \text{ and } y _ 2 = -\frac{1+\sqrt{3}}{2}[/tex]
Step-by-step explanation:
We are given the equation:
[tex]\displaystyle (x^2 + 2xy) + i(y-1) = (x^2 -2x + 2y) - i(x +y)[/tex]
And we want to find the values of x and y such that the equation is true.
First, distribute:
[tex]\displaystyle (x^2 + 2xy) + i(y-1) = (x^2 -2x + 2y) +i(-x -y)[/tex]
If two complex numbers are equivalent, their real and imaginary parts are equivalent. Hence:
[tex]\displaystyle x^2 + 2xy = x^2 - 2x +2y \text{ and } y - 1 = -x -y[/tex]
Simplify:
[tex]\displaystyle 2xy = -2x +2y \text{ and }x = 1 - 2y[/tex]
Substitute:
[tex]\displaystyle 2(1-2y)y = -2(1-2y) + 2y[/tex]
Solve for y:
[tex]\displaystyle \begin{aligned} 2(y - 2y^2) &= (-2 + 4y) + 2y \\ 2y - 4y^2 &= 6y -2\\ 4y^2 + 4y - 2& = 0 \\ 2y^2 + 2y - 1 &= 0 \\ \end{aligned}[/tex]
From the quadratic formula:
[tex]\displaystyle \begin{aligned} y &= \frac{-(2)\pm\sqrt{(2)^2 - 4(2)(-1)}}{2(2)} \\ \\ &= \frac{-2\pm\sqrt{12}}{4} \\ \\ &= \frac{-2\pm2\sqrt{3}}{4}\\ \\ &= \frac{-1\pm\sqrt{3}}{2} \end{aligned}[/tex]
Hence:
[tex]\displaystyle y_1 = \frac{-1+\sqrt{3}}{2} \text{ or } y_2 = \frac{-1-\sqrt{3}}{2}[/tex]
Then:
[tex]\displaystyle x _ 1 = 1 - 2\left(\frac{-1+\sqrt{3}}{2}\right) = 1 + (1 - \sqrt{3}) = 2 - \sqrt{3}[/tex]
And:
[tex]\displaystyle x _ 2 = 1 - 2\left(\frac{-1-\sqrt{3}}{2}\right) = 1 + (1 + \sqrt{3}) = 2 + \sqrt{3}[/tex]
In conclusion, the values of x and y are:
[tex]\displaystyle x_1 = 2-\sqrt{3} \text{ and } y_1 = \frac{\sqrt{3}-1}{2}[/tex]
Or:
[tex]\displaystyle x _ 2 = 2 + \sqrt{3} \text{ and } y _ 2 = -\frac{1+\sqrt{3}}{2}[/tex]
Examine the tile pattern at right
Answers
b. The pattern grow by adding 1 tile above the tile and adding 1 tile at the right of the tile.
c. In figure 0, there will be 1 tile. We know this because in each successive figures a tile is added at the above and a tile is added to the right, so ineach preceeding figure the same is reduced. In figure 1, there are e tiles, so in figure 0, there will be 3-2 = 1 tile.
20. In the above figure, ZAOB = 80°. What does ZACB equal?
A. 10°
B. 160°
C. 80°
O
D. 40°
Answers
Answer:
D. 40
Step-by-step explanation:
A central angle is equal to the measure of its corresponding arc.
angle AOB is a central angle and it's corresponding arc is arc AB
this means that arc AB = measure of angle AOB
If angle AOB = 80 degrees then arc AB equals 80 degrees too.
An inscribed angle is equal to half the measure of its intercepted arc.
Angle ACB is an inscribed angle and the arc it intercepts is arc AB
This means that angle ACB = 1/2 of arc AB
We have found that the measure of arc AB is 80 degrees.
This means that angle ACB = 1/2 of 80 which is 40
Mrs. Diaz had 5/6 gallon of paint to start. When she finished sha had 1/2 gallon. How much paint did Mrs. Diaz use?
Answers
Answer:
1/3
Step-by-step explanation:
To get the amount of paint,you will have to deduct the number of paint she used to start minus the amount of of gallon she finished with.
5/6-1/2
L.C.M
10-6/12
4/12
=1/3
The bases of a prism are always
A. perpendicular
B. parallel
C. intersecting
D. rectangles
Answers
Answer:
D. rectangles
Step-by-step explanation:
Perpendicular is when they cross but the base isn't perpendicular
Parallel is when two lines never touch, but in the base of a prism there are lines that touch
Intersecting is when two or more lines meet and cross over each other
please help A data set has its first and third quartiles as 9 and 17, respectively.
Which of the following data points would be considered an outlier for the data set?
27
17
3
41
Answers
Answer: 27
Step-by-step explanation: hihihihihiiiiihihi
27 is considered an outlier for the data set.
What is quartiles?In statistics, a quartile exists as a variety of quantiles which divides the number of data points into four parts, or quarters, of more or less equal size. The data must be ordered from smallest to largest to calculate quartiles; as such, quartiles exist a form of order statistic.
A quartile separates data into three points—a lower quartile, median, and upper quartile—to form four sets of the dataset.
Hence, 27 is considered an outlier for the data set.
To learn more about quartiles refer to:
https://brainly.com/question/9260741
#SPJ2
Find any domain restrictions on the given rational equation x/x+4 + 12/x^2+5x+4 = 8x/5x-15
Answers
Answer:
x ≠ -4, -1, 3
Step-by-step explanation:
12/(x^2+5x+4) = 8x/(5x-15)
12/((x+1)(x+4)) = 8x/(5(x-3))
division by zero is undefined
There are 3 answers to your question. x= -1 x= 3 x= -4
The winning team's score in 5 high school basketball games was recorded. If the sample mean is 62.3 points and the sample standard deviation is 11.0 points, find the
98% confidence interval of the true mean.
A) 57.4 < µ < 67.2 B) 50.8 < µ < 73.8
C) 25.4 < µ < 99.2 D) 43.9 < µ < 80.7
Answers
Answer:
This is 3.747*11/sqrt(5)=18.4
the interval is (49.9,86.7), the SE added to and subtracted from the mean.
It is B
Thank You
Find the place value of 7 in 374596
Answers
Answer:
It is tens
Step-by-step explanation:
3 » ones
[tex]{ \boxed{ \sf{7 \: » \: tens}}}[/tex]
4 » hundreds
5 » thousands
9 » ten thousands
6 » hundred thousands
Rational or Irrational: -4 √81
Answers
Answer:
it is rational
Step-by-step explanation:
Answer:
it's rational number
I hope it's helps you
-1 and 3 tenths plus 4 tenths
Answers
The answer you are looking for is -9/10, or -0.9.
Solution:
-1 3/10+4/10
-13/10+4/10
-9/10, or -0.9
Answer:
-0.9
Step-by-step explanation:
cho hàm số y= x^{3} +3x^{2} -1 .viết phương trình tiếp tuyến tại điểm có hoành độ bằng -1
Answers
hope you do not mind expressing this on in English ♂️
In a survey of small business owners, a response to which of the following questions would be qualitative?
a. How much did your business have in profits last year?
b. How long have you owned a business?
c. What type of business do you own?
d. How many businesses do you own?
Answers
For the considered survey, the questions whose response is going to be qualitative is given by: Option C: "What type of business do you own?"
What is qualitative data?
Qualitative data tells about the quality or characteristic. It is tough to express it numerically or not at all expressible numerically. They are usually catagorical.
In contrast, there is quantitative data which can be expressed numerically.
Option A, Option B and Option D will get answers which would be in numbers. Thus, their responses would be quantitative.
Option C's question will get response as type of business that the business owners own. So its not in numbers and tells about the characteristic of businesses.
Thus, for the considered survey, the questions whose response is going to be qualitative is given by: Option C: "What type of business do you own?"
Learn more about qualitative and quantitative data here:
https://brainly.com/question/12929865
find the GCF from the two numbers and rewrite the sum using nthe distributive property
24 + 36
Answers
Answer:
The greatest common factor is 6.
Step-by-step explanation:
Greatest common factor is 6. If you use the distributive property then the answer would be 6(4) + 6(6) or 6(4+6). Then you distribute the 6 to each digit and should get 24+36.
