The property used in this problem are
commutative property of additionassociative property of additionadditive inverseadditive identityProperty of numbers.
We know that, there are four basic properties of numbers.
They are commutative, associative, distributive, and identity.
Given,
Here we have the problem
[38 + 677] + (-38)
[677 + 38] + (-38)
677 + [38 + (-38)]
677 + 0
677
Now, we need to identify all properties that were used to simplify the problem.
In the first step of the problem is,
[38 + 677] + (-38)
We have used the commutative property of addition to interchange the given numbers in the brackets,
[677 + 38] + (-38)
Now, we have to use the associative property of addition to group the numbers,
677 + [38 + (-38)]
Then we have to use the additive inverse, to get the value of brackets,
677 + 0
Finally we have to use the additive identity to get the result.
677
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Internet rates for two companies are shown in the table. After how many months of service is the total cost thesame no matter which company is used? What is this total cost?Intemet Service ProviderChargesTyson's Ethernet $100 activation fee plus $20 per monthDarriana's DSL$30 per month
We need to find in which month we have that the total cost is the same, so we are going to do the equations of cost of the two companies
For the Tyson's Ethernet we have that the equation is
[tex]f(m)=20\cdot m+100[/tex]with m the month. And for the Darriana's DSL the equation is
[tex]g(m)=30\cdot m[/tex]We need to equalize the equations, so we have
[tex]\begin{gathered} 30m=20m+100 \\ 30m-20m=20m+100-20m \\ 10m=100 \\ m=\frac{100}{10}=10 \end{gathered}[/tex]Now to see the cost we replace in the two equations( wiith this we also verify), we have that
[tex]\begin{gathered} f(10)=20\cdot10+100=200+100=300 \\ g(10)=30\cdot10=300 \end{gathered}[/tex]The answer is: after 10 months the total cost is the same, and this cost is $300.
Look at the four company logos below.VolkswagenLincolnLexusRed Cross0♡+The logo for Volkswagen haslines of symmetry.The logo for Lincoln haslines of symmetry.The logo for Lexus haslines of symmetry.The logo for Red Cross has4lines of symmetry.:: 0.: 1:: 2
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
Figures logos
Lines of symmetry = ?
Step 02:
We must analyze the logos to find the solution.
Volkswagen ===> 1 line of symmetry
Lincoln ===> 2 lines of symmetry
Lexus ===> 0 lines of symmetry
Red Cross ===> 4 lines of symmetry
That is the solution.
Answer:
VW: 1
Lincoln: 2
Lexus: 0
Red Cross: 2
Step-by-step explanation:
VW can be split in half once, and have the same thing son both sides.
Lincoln be split in half horizontally and vertically and be identical on both sides
Lexus cannot be split in half at all and be identical.
The red cross can be split horizontally and vertically and still have identical pieces.
-Hope this helped
The arcade charges $125.00 to reserve the location, and then $15.00 per person. Which expressionrepresents the total cost for any number of people n?
Let 'n' represent the number of people. Sin the arcade charces 15 per person, then we have the following first term:
[tex]15n[/tex]since the charge to reserve is $125, we have:
[tex]15n+125[/tex]thus, if 'c' represents the total cost, then the expression to represent this situation is:
[tex]c=15n+125[/tex]Complete the synthetic division work below to divide 5x^3 - 3x^2 + 2x - 1 by x+1 Fill in the blank with the four numbers that would go below the line. Separate numbers by commas. 5 -3 2 - 1
Abraham, this is the solution of the polynomials division:
5x^3 - 3x^2 + 2x - 1/ x+1
Step 1:
5x² + (-8x² + 2x - 1)/(x + 1)
Step 2:
5x² - 8x + (10x - 1)/(x + 1)
Step 3:
5x² -8x + 10 + (-11/x + 1)
The four numbers below the line are: 5 -8 10 - 11
Hi I need help with this question. Melissa starts with four packages of balloons and 8 single balloons. Her brother gave her 20 more balloons and three more packages. She now has twice the number of balloons she started with. Write and solve an equation to find how many balloons are in a package. How many balloons did Melissa start with?
Okay, here we have this:
Let's take the packages as (x), so we obtain:
Melissa Starts with: 4x+8 ballons (1)
Her brother gave her: 3x+20 ballons (2)
And as she now has twice the number of balloons she started with. This mean that:
2(4x+8)=(3x+20)+(4x+8)
Let's clear x in this equation:
8x+16=3x+20+4x+8
8x+16=7x+28
8x-7x=28-16
x=12
Finally we obtain that in a package are 12 ballons, now let's calculate How many balloons did Melisa start with?:
Remember that Melissa Starts with: 4x+8 ballons, so replacing x with 12, we obtain that Melissa Starts with: 4(12)+8=48+8=56.
Finally we obtain that Melissa Starts with 56 ballons.
16. The table below shows the population of California from 2010 to 2019.
The Solution:
The Regression Model that best fits the data given in the question is
[tex]P(t)=\frac{a}{1+e^{-bt}}[/tex]From the Desmos plotter analysis attached above, we have that
[tex]\begin{gathered} a=74.907\approx74.91 \\ b=0.01397\approx0.01 \end{gathered}[/tex]So, by substituting the values of the parameters, we get the required logistic Regression Model is given below:
[tex]P(t)=\frac{74.91}{1+e^{-0.01t}}[/tex]b. The model predicts that the population of California in 2025 will be:
[tex]\begin{gathered} \text{From 2010 to 2025 is 25 years.} \\ So,\text{ t=25 years.} \\ S\text{ubstituting 25 for t in the regression model, we get} \end{gathered}[/tex][tex]P(t)=\frac{74.91}{1+e^{-0.01(25)}}=\frac{74.91}{1+0.77088}=\frac{74.91}{1.77088}=42.1127\approx42.1\text{ million people.}[/tex]So, the population of California in 2025 will be 42.1 million people.
c. To find when the model predicts that the population of California will be 40 million,
we shall substitute 40 (in millions) for P in the model, and find t as below:
[tex]40=\frac{74.91}{1+e^{-0.01t}}[/tex]Cross multiplying, we get
[tex]\begin{gathered} 40(1+e^{-0.01t})=74.91 \\ \text{Dividing both sides by 40},\text{ we have} \\ 1+e^{-0.01t}=\frac{74.91}{40} \\ \\ 1+e^{-0.01t}=1.87275 \\ e^{-0.01t}=1.87275-1 \\ e^{-0.01t}=0.87275 \end{gathered}[/tex]Taking the ln of both sides, we get
[tex]\begin{gathered} \ln e^{-0.01t}=\ln 0.87275 \\ -0.01t=\ln 0.87275 \\ -0.01t=-0.136106 \\ \text{ Dividing both sides by -0.01, we get} \\ t=\frac{-0.136106}{-0.01}=13.61\approx14\text{ years} \end{gathered}[/tex]So, 14 years from 2010 will be in the year 2024
d. According to the model, the carrying capacity for California's capacity is 74.9 million people.
write a quadratic function whose graph has the given characteristicsvertex: (1,2)Point: (3,6)
Solution
For this case the general expression for a parabola is given by:
y- k= a(x-h)^2
From the info given we know that h = 1 and k= 2
And replacing we got:
y -2 = a(x-1)^2
And replacing the point given we got:
6-2 = a(3-1)^2
4= 4a
a=1
And the equation would be given by:
y-2 = (x-1)^2
10. f(x) = 2x+5 if x < 4 1x² + 3x if x 24 LX D = R =
The domain of the given piecewise function are all the values that x can take.
These are:
x < -4 and
[tex]x\ge-4[/tex]It can be written as:
[tex](-\infty,\infty)[/tex]The range is:
[tex](-\infty,-3)\cup\lbrack-\frac{9}{4},\infty)[/tex]Please I need help on this ASAP
The value of the function h(x) in part (a) is equal to h(x) = 2x.
The value of the function h(x) in part (b) is equal to h(x) = 5x/2 - 2.
How to determine the function h(x)?In this exercise, you are required to calculate the value of the function h(x), which is a product of the addition of function f(x) and function g(x). This ultimately implies that, the value of function h(x) can be calculated by adding function f(x) and function g(x) together.
For the first part (a), the value of function h(x) would be calculated as follows:
Function h(x) = Function f(x) + Function g(x)
Substituting the given parameters into the formula, we have;
Function h(x) = (x + 4) + (x - 4)
Function h(x) = x + 4 + x - 4
Function h(x) = 2x
For the second part (b), the value of function h(x) would be calculated as follows:
Function h(x) = Function f(x) + Function g(x)
Substituting the given parameters into the formula, we have;
Function h(x) = (2x - 4) + (x/2 + 2)
Function h(x) = 2x - 4 + x/2 + 2
Function h(x) = 5x/2 - 2.
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How many distinct rearrangements of the letters in 'PUYGPGPYYUG' are there?
The answer is 92400
To solve this, we can count how many letters we have. There are 11 letters.
If those 11 letters were different from each other, the answer would be 11!
But we have letters that repeats:
3 P's
3 Y'2
3 G'2
2 U's
Since we want to know the quantity of distinct arrangements, we can divide by the repetition. This means:
[tex]\begin{gathered} 11!\text{ = total combinations} \\ \frac{11!}{3!\cdot3!\cdot3!\cdot2!}=\text{total distinct combinations} \end{gathered}[/tex]We divide by 3! times 3! times 3! times 2!, because we have 3P's, 3Y's, 3G's and 2U's
Then on the calculator write the division and give us the answer 92400
Woad holder mange en windows-17Grado2rean, thetwo w Wawie kan 137( wow month am wa mama w War[mmer
a) The temperature at Milwaukee by 6am is given to be -5 degree fahrenheit.
By noon, this temperature has risen to 13 degree fahrenheit, thus this implies;
[tex]-5+13=8^0F[/tex]Hence, the temperature at Milwaukee by noon will be 8 degree fahrenheit
b) Temperature in Winnipeg = -17 degree fahrenheit
Temperature in Orlando = 61 degree fahrenheit
Temperature in Winnipeg is lower than in Orlando implies;
[tex]61-(-17)=61+17=78^0F[/tex]Hence, the temperature in Winnipeg is 78 degree fahrenheit lower than in Orlando
Answer the statistical measures and create a box and whiskers plot for the followingset of data.4,4,5,5,5,6,6, 8, 9, 10, 12, 14, 14, 15, 17
Notice that the set of data is already ordered from lowest to highest, therefore the minimum value is 4 and the maximum value is 17.
The median is 8, the first quartile Q₁=5, and Q₃=14.
Then, the corresponding box and whiskers plot is:
Hi this is one of my exit ticket problem can you solve it please.
The triangles that are similar is A and B
if andrew can run 60 meters in 6 seconds, how many meters can he run in 1 second?
Explanation
Step 1
the sp
Find the prime factorization of 54 from least to greatest; then write it using exponents.
The prime factorization of 54 is
54 = 2 × 3 × 3 ×3
Using the exponential form of the prime factorization that is
54 = 2 × 3³
Lily drank 2 1/2 cartons of juice in the month of January. In the month of February, she drank twice as many cartons of juice as in January. How many cartons of juice did she drink in February?
Answer:
Lily drank 5 cartons of juice in February.
Explanation:
From the question, we're told that in January, Lily drank 2 1/2 cartons of juice and drank twice of that amount in February, so all we need to do is multiply the amount she drank in January by 2 to get the amount she drank in February.
Solving this, we'll have;
[tex]2\ast(2\frac{1}{2})=2\ast(\frac{5}{2})=5[/tex]Therefore, Lily drank 5 cartons of juice in February.
Use the 'Permutations' formula to evaluate the expression P(27,3)
The permutation formula is given as:
[tex]P(n,k)=\frac{n!}{(n-k)!}[/tex]In this case n=27 and k=3; plugging the values in the formula we have:
[tex]P(27,3)=\frac{27!}{(27-3)!}=\frac{27!}{24!}=\frac{27\cdot26\cdot25\cdot24!}{24!}=27\cdot26\cdot25=17550[/tex]Therefore, we have that P(27,3)=17550
Martha opened a savings account and deposited 400.00 the account earns 1%interest compounded annually what is the balance after 3 years
p = 400.00
r = 1% = 1/100 = 0.01
t = 3 years
Therefore,
[tex]\begin{gathered} Amount=p(1+\frac{r}{n})^{nt} \\ \text{Amount}=400(1+\frac{0.01}{1})^{1\times3} \\ \text{Amount}=400\times1.030301 \\ \text{Amount}=\text{ \$}412.1204 \\ \text{Amount}=\text{ \$412.12} \end{gathered}[/tex]After three years the amount will be = $412.12
Use elimination to solve each system of equations.4x + y = 233x - y = 12Use elimination to solve each system of equations.4x + y = 233x - y = 12
ANSWER
The solution is (5, 3)
EXPLANATION
To use elimination method, we have to subtract (or add) one equation from the other, in order to eliminate one of the variables. Then we'll have one equation with one variable. We solve it for that variable and then replace into one of the equations of the system to solve for the eliminated variable.
In these equations, we have +y in the first one and -y in the second one. We can add both equations to eliminate y:
[tex]\begin{gathered} (4x+y)+(3x-y)=23+12 \\ 4x+3x+y-y=35 \\ 7x=35 \end{gathered}[/tex]Solving for x:
[tex]\begin{gathered} 7x=35 \\ x=\frac{35}{7} \\ x=5 \end{gathered}[/tex]Now, we have to replace x = 5 into one of the equations and solve for y. Replacing in the first equation:
[tex]\begin{gathered} 4x+y=23 \\ 4\cdot5+y=23 \\ 20+y=23 \\ y=23-20 \\ y=3 \end{gathered}[/tex]The solution to this system is (5, 3)
Find from first principles the derivative of f(x)= root of X with respect to x
To find:
The derivative of function f(x) using the first principle.
[tex]f(x)=\sqrt{x}[/tex]Solution:
By the first principle, the derivative of the function f(x) is given by:
[tex]f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]So, the derivative of the given function can be obtained as follows:
[tex]\begin{gathered} f^{\prime}(x)=\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h} \\ =\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\times\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}} \\ =\lim_{h\to0}\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})} \\ =\lim_{h\to0}\frac{h}{h(\sqrt{x+h}+\sqrt{x})} \\ =\lim_{h\to0}\frac{1}{(\sqrt{x+h}+\sqrt{x})} \\ =\frac{1}{\sqrt{x+0}+\sqrt{x}} \\ =\frac{1}{2\sqrt{x}} \end{gathered}[/tex]Thus, the derivative of the given function is:
[tex]f^{\prime}(x)=\frac{1}{2\sqrt{x}}[/tex]Alex has a box of chocolates which are all
flavoured with either caramel or raspberry.
The possible outcomes of Alex picking two
chocolates at random are shown in the tree
diagram below.
If both chocolates are the same flavour, what
is the probability that they are both
raspberry?
Give your answer as a fraction in its simplest
form.
Probability that they are both raspberry = 5/33
From the question, we have
probability that they are both raspberry = 5/12*4/11
=5/33
Probability:
Possibility is referred to as probability. This branch of mathematics deals with the occurrence of a random event. The value's range is 0 to 1. Probability has been applied into mathematics to predict the likelihood of different events. Probability generally refers to the degree to which something is likely to occur. This fundamental theory of probability, which also applies to the probability distribution, can help you comprehend the possible outcomes for a random experiment. Gonna determine how likely something is to occur, use probability. Many things are hard to predict with 100% certainty. We can only anticipate the possibility of an event occurring using it, or how likely it is.
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The shorter leg of a 30°-60°-90° triangle measures 9sqrt3 inches. What is the length of the longer leg? OA. 27 inches OB. 27sqrt3 inches OC. 18 inches OD. 18sqrt3 inches
We know that the proportion of the sides of a 30°-60°-90° triangle is:
The shorter leg is K, then:
[tex]K=9\sqrt[]{3}\text{ in}[/tex]Using this result, we can calculate the length of the longer leg:
[tex]\begin{gathered} \sqrt[]{3}K=\sqrt[]{3}\cdot9\cdot\sqrt[]{3}=9\cdot3 \\ \Rightarrow=27\text{ in} \end{gathered}[/tex]Santa is saving money for a new bike that costs $175.62. He already has $39.59. How much more does he need to save before he can buy a bike?
Given
Santa is saving money for a new bike that costs $175.62. He already has $39.59.
Answer
Total money needed = 175.62 - 39.59 = $136.03
second blank has the option of , the same verticle asymptote as function h, vertical asymptote at x=-7, vertical asymptote at x=-5, and vertical asymptote at x=3
Given:
The graph is g(x) is given and the function h(x) is,
[tex]h(x)=g(x+5)[/tex]To classify the asymptotes:
Since the translated transformation of 5 units left,
There are no changes in the horizontal asymptote.
But, the vertical asymptote is,
[tex]\begin{gathered} x=-2-5 \\ =-7 \end{gathered}[/tex]Thus, the graph h(x)=g(x+5) has the same horizontal asymptote as the function g
the answer and how to figure questions out like this!
In order to find the exponential regression we are going to select some values of the given data.
STEP 1An special value is when x=0.
On the table we can see that when x=0 then y=9
Replacing x by 0 in the given choices, we have that:
[tex]\begin{gathered} A\text{.} \\ y=8.04\cdot0.98^x \\ \downarrow \\ y=8.04\cdot0.98^0=8.04\cdot1 \\ =8.04 \end{gathered}[/tex][tex]\begin{gathered} B\text{.} \\ y=3.02\cdot3.67^x \\ \downarrow \\ y=3.02\cdot3.67^0=3.02\cdot1 \\ =3.02 \end{gathered}[/tex][tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^0=6.61\cdot1 \\ =6.61 \end{gathered}[/tex][tex]\begin{gathered} D\text{.} \\ y=2.27\cdot2.09^x \\ \downarrow \\ y=2.27\cdot2.09^0=2.27\cdot1 \\ =2.27 \end{gathered}[/tex]Observing the results we have that the two choices with closer results to 9 are A (with 8.04) and C (with 6.61)
STEP 2Now, we are going to select two additional values from the table in order to find which is the best answer: A or C.
Let's take x=1.
When x = 1, then y=10.
Replacing on the equation A we have:
[tex]\begin{gathered} A\text{.} \\ y=8.04\cdot0.98^x \\ \downarrow \\ y=8.04\cdot0.98^1=8.04\cdot0.98 \\ =7.879 \end{gathered}[/tex]and for the equation C:
[tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^1=6.61\cdot1.55 \\ =10.2455 \end{gathered}[/tex]For x=1, the nearest result is from the equation C.
Let's verify what happens when x=2.
When x=2 then y=16. Replacing on the equation A we have:
[tex]\begin{gathered} A\text{.} \\ y=8.04\cdot0.98^x \\ \downarrow \\ y=8.04\cdot0.98^2 \\ =7.7216 \end{gathered}[/tex]and for the equation C:
[tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^2 \\ =15.88 \end{gathered}[/tex]Again, for x=2, the nearest result is from the equation C.
Then, we can conclude that the best candidate is equation C.
We could try other values of x to double check:
[tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^2 \\ =15.88 \end{gathered}[/tex]I don't know if -32 is greater or less than 15
Answer: -32 is less than 15.
Step-by-step explanation: 15 is greater than zero therefore it is not a negative number. -32 is less than 0 therefore it is a negative number. Negatives will always be less than positives.
If F(x) =8+11 X-3X², finda. What is F (5)? b. What is F (x + b)?c. What is F (-3)?
a. Solve for F(5).
To solve for F(5), substitute x = 5 to the given function, and evaluate accordingly to the operation
[tex]\begin{gathered} F(x)=8+11x-3x^2 \\ F(5)=8+11(5)-3(5)^2 \\ F(5)=8+55-3(25) \\ F(5)=63-75 \\ F(5)=-12 \end{gathered}[/tex]b. Solve for F(x+b)
Again we substitute the paremeters by x+b to the given function, and we get
[tex]\begin{gathered} F(x)=8+11x-3x^2 \\ F(x+b)=8+11(x+b)-3(x+b)^2 \\ F(x+b)=8+11x+11b-3(x^2+2xb+b^2) \\ F(x+b)=8+11x+11b-3x^2-6xb-3b^2 \\ \\ \text{Since we cannot simplify further, we just rearrange} \\ \text{the final answer according to the degree of the terms} \\ F(x+b)=-3x^2-3b^2-6xb+11x+11b+8 \end{gathered}[/tex]c. Solve for F(-3)
Substitute x = -3
[tex]\begin{gathered} F(x)=8+11x-3x^2 \\ F(-3)=8+11(-3)-3(-3)^2 \\ F(-3)=8-33-3(9) \\ F(-3)=-25-27 \\ F(-3)=-52 \end{gathered}[/tex]Select the following sentence that represents the equation below:
3(n2+1)=3n+12
Responses
The sum of the product of two and a number plus one times three is equal to twelve more than the product of three and the same number.
The sum of the product of two and a number plus one times three is equal to twelve more than the product of three and the same number., EndFragment,
Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number.
Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number., EndFragment,
The quotient of a number and two increased by one is equal to twelve more than the quotient of three and the same number.
The quotient of a number and two increased by one is equal to twelve more than the quotient of three and the same number., EndFragment,
Three times the sum of a number divided by two and one is equal to three times the same number increased by twelv
The answer is, Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number.
Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number.
What is a quotient?It has a wide spread throughout the Mathematics. It is referred to as the integer part of a division, or as the fraction, or as a ratio. It is used when indicating the presence or the degree of a characteristic in something or by someone. Quotient is the result of a division. It is obtained when we divide one number by another. Quotient means how many times. And it is derived from the Latin.
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Question #2Several points are plotted on the graph. Which of the plotted points on the graph represent the zeros of the function:f (x) = (x^2+ 2x- 8) (x – 6)
Given function is:
[tex]f(x)=(x^2+2x-8)(x-6)[/tex]Now put f(x)=0
[tex]\begin{gathered} (x^2+2x-8)(x-6)=0 \\ (x^2+2x-8)=0,(x-6)=0 \\ x-6=0 \\ x=6 \end{gathered}[/tex]so the first point is (6,0) now for another points:
[tex]\begin{gathered} (x^2+2x-8)=0 \\ (x^2+4x-2x-8)=0 \\ x(x+4)-2(x+4)=0 \\ (x+4)(x-2)=0 \\ x=-4,2 \end{gathered}[/tex]So the points are (-4,0) and (2,0) and(6,0)
Hence the correct options are 1,2 and 4.
identify the reflection of the figure with vertices P (2, -12), Q (-3, 13), and R (-5, - 15) across the x-axis.
EXPLANATION:
We are given the following coordinates for a figure on the coordinate plane;
[tex]\begin{gathered} P(2,-12) \\ Q(-3,13) \\ R(-5,-15) \end{gathered}[/tex]To reflect any figure or any set of coordinates across the x-axis, we shall apply the rule;
[tex](x,y)\rightarrow(x,-y)[/tex]Note that the x-coordinate remains the same whereas, the y-coordinate changes its sign.
Imagine folding a graph page in two equal halves along the horizontal axis. You'll observe the y-coordinates will switch sides from top to bottom (positive to negative) or bottom to top (negative to positive). The x-coordinate remains the same since moving the folded page does not affect values along the horizontal line.
Therefore, for the vertices given, the reflection across the x-axis would be;
[tex]P(2,-12)\rightarrow P^{\prime}(2,12)[/tex][tex]Q(-3,13)\rightarrow Q^{\prime}(-3,-13)[/tex][tex]R(-5,-15)\rightarrow R^{\prime}(-5,15)[/tex]ANSWER:
The coordinates of the reflection across the x-axis therefore will be;
[tex]\begin{gathered} P^{\prime}(2,12) \\ Q^{\prime}(-3,-13) \\ R^{\prime}(-5,15) \end{gathered}[/tex]