For the following function, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then, graph the function and state the domain and the vertical asymptote. f(x) = 7 - In x Describe how the graph of f(x) can be obtained from the graph of a basic logarithmic function. The graph of f(x) = 7 - In x is a transformation of the graph of f(x) = In x by a reflection across the and then a translation units. Use the graphing tool to graph the equation.
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Answer
1) Graph is shown below in the 'Explanation'.
2) Domain: x > 0
In interval notation,
Domain: (0, ∞)
3) Vertical asymptote: x = 0
Horizontal asymptote: y = 7
4) The transformations required to turn f(x) = In x into f(x) = 7 - In x include
A reflection of f(x) = In x about the x-axis.
Then, this reflected image is then translated 7 units upwards.
Explanation
The graph of function is attached below
For the domain and asymptote,
Domain
The domain of a function refers to the values of the independent variable (x), where the dependent variable [y or f(x)] or the function has a corresponding real value. The domain is simply the values of x for which the output also exists. It is the region around the x-axis that the graph of the function spans.
We know that the logarithm of a number only exists if the number is positive.
So,
Domain: x > 0
In interval notation,
Domain: (0, ∞)
Asymptote
Asymptotes are the points on either the x-axis or the y-axis where the graph of the function doesn't touch.
They are usually denoted by broken lines.
For this question, we know that the value of f(x) cannot go beyond f(x) = 7 and x = 0
Vertical asymptote: x = 0
Horizontal asymptote: y = 7
For the transformation
When a function f(x) is translated horizontally along the x-axis by a units, the new function is represented as
f(x + a) when the translation is by a units to the left.
f(x - a) when the translation is by a units to the right.
When a function f(x) is translated vertically along the y-axis by b units, the new function is represented as
f(x) + b when the translation is by b units upwards.
f(x) - b when the translation is by b units downwards.
So, if the original function is
f(x) = In x
f(x) = -In x
This reflects the original function about the x-axis.
Then,
f(x) = 7 - In x
This translates the reflected function by 7 units upwards.
Related Questions
Three lakes lost water during a drought. Lake Jensen lost one ninth of its water, Lake Parlow lost 10% of its water, and Lake Stockton lost twenty one two hundredths of its water. Which lake lost the least amount of water?
Lake Jensen
Lake Parlow
Lake Stockton
All three lakes lost the same amount of water.
Answers
The lake that lost the least amount of water during the drought is Lake Parlow.
Which lake lost the least amount of water?The amount of water lost by Lake Jensen is written as a fraction. A fraction is a non-integer that is made up of a numerator and a denominator. An example of a fraction is 1/8.
The amount of water lost by Lake Parlow is written as a percentage. Percentage is the fraction of an amount that is written as a number out of 100.
The amount of water lost by Lake Stockton is written as a decimal. A decimal is the standard form used to write non-integers.
twenty one two hundredths = 21 / 200 = 0.105.
In order to compare the water lost by the three lakes, convert the fraction and the decimal to percentage.
1/9 x 100 = 11.11%
0.105 x 100 = 10.5%
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give a reason that justifies each statement for questions 4-9.
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4.
The angles ∠6 and ∠8 are congruent because they are alternate interior angles.
5.
The angles ∠4 and ∠5 added are equal to 180° because they are supplementary angles.
6.
If ∠1 is equal ∠7, then they are alternate exterior angles, therefore the lines p and q need to be parallel.
7.
The angles ∠1 and ∠2 are congruent because they are vertically opposite angles.
8.
If ∠2 and ∠6 are supplementary, they are consecutive interior angles, therefore the lines p and q need to be parallel.
9.
The angles ∠6 and ∠9 are congruent because they are corresponding angles.
Express 2x-3y=-6 into y=mx+b
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For this problem, we are given a certain expression and we need to write it in the "y=mx+b" form.
We need to isolate the "y" variable on the left side to solve this problem. We have:
[tex]\begin{gathered} 2x-3y=-6\\ \\ 2x-3y-2x=-6-2x\\ \\ \frac{-3y}{-3}=\frac{-6}{-3}-\frac{2x}{-3}\\ \\ y=2+\frac{2}{3}x\\ \\ y=\frac{2}{3}x+2 \\ \\ \end{gathered}[/tex]The expression is y = (2/3)x+2.
16. Given the graph below, write the equation of the line graphed. Equation:
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We have the following:
The equation has the following form
[tex]y=mx+b[/tex]where m is the slope and b is y-intercept
The slope formula is as follows
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]The point are (-4, 8) and (6, -4)
replacing:
[tex]m=\frac{-4-8}{6-(-4)}=\frac{-12}{10}=-\frac{6}{5}[/tex]In the graph we can see that the y-intercept is equal to 4, therefore, the equation would be
[tex]y=-\frac{6}{5}x+4[/tex]I need geometry help please.
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Lets remember the property "alternative-interior angles" when we have parallels lines:
Given two paralles lines, the following is true:
In our question, we have:
We know that the angles J + P + K=180
So, the triangles have all the angles the same, so they are similar by angle-angle-angle, and they have one side congruent, AC=CD, so the triangles are congruents.
Solve for a: a = 10.
a = , ,
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Answer:
a=10 and a=-10
Step-by-step explanation: its absolute value
in 3 years Donald wants to buy a bicycle that costs 600.00 if he opens a savings account that earns 4% interest compounded quarterly how much will he have to despoit as principal to have enough money in 3 years to buy the bike
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We want the future value to be $600. With an interest of 4% quarterly in 3 years, we have the following information:
[tex]\begin{gathered} FV=600 \\ i=0.04 \\ t=3 \\ n=4 \end{gathered}[/tex]Then we apply the following formula:
[tex]PV=\frac{FV}{(1+\frac{i}{n})^{n\cdot t}}[/tex]therefore, we have that:
[tex]PV=\frac{600}{(1+\frac{0.04}{4})^{4\cdot3}}=\frac{600}{(1.01)^{12}}=532.46[/tex]therefore, Donald would have to deposit $532.46 as principal.
Find the value of x to make this equation true. 6x + 1 = 6 + 2x.
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6x + 1 = 6 + 2x.
collect the like term
6x - 2x = 6-1
4x = 5
divide both-side of the equation by 4
4x/4 = 5/4
x=1.25
1.25
Construct a probability distribution for a discrete random variable uses the probability experiment of tossing a coin three times. Consider the random variable for the number of heads
Answers
Answer:
Explanation:
By building a tree diagram we can find the theoretical probability of each number of heads when tossing three coins.
What is the slope of a line perpendicular to the line whose equation is 3x-5y=45. Fully simplify your answer
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The slope of a line perpendicular to the line whose equation is 3x-5y=45 is -5/3.
So first of all, we have to find the slope of the given line. Convert it into Slope-Intercept Form.
The Slope - Intercept Form is : y = mx + c
Converting the given equation, we get :
3x - 5y = 45
5y = 3x + 45
y = (3/5)x + 15
Perpendicular Lines
The lines having opposite reciprocal slopes are perpendicular. That means you flip the sign (+/-) and flip the numerator and denominator. The slope of the line perpendicular to this one is -5/3.
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can you please solve this practice problem for me I need assistance
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The missing angle in the triangle of the left is:
51 + 74 + x = 180
x = 180 - 51 - 74
x = 55°
The missing angle in the triangle of the right is:
55 + 74 + x = 180
x = 180 - 55 - 74
x = 51°
Then, both triangles are similar. This means that their corresponding sides are in proportion. These sides are:
35 in
All of the following ratios are equivalent except 8 to 12 15/102/36:9
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False
1) Let's examine those ratios, and simplify them whenever possible:
[tex]\begin{gathered} \frac{15}{10}=\frac{3}{2} \\ \frac{2}{3} \\ \frac{6}{9}=\frac{2}{3} \\ \frac{8}{12}=\frac{2}{3} \end{gathered}[/tex]2) Simplifying those ratios, all the following but 15/10 are equivalent to 8/12
3) So this is a false statement to say that all of those are equivalent except 8 to 12.
At a college basketball game, the ratio of the number of freshmen who attended to the number of juniors who attended is 3:4. The ratio of the number of juniors who attended to the number of seniors who attended is 7:6. What is the ratio of the number of freshmen to the number of seniors who attended the basketball game?
A) 7:8
B) 3:4
C) 2:3
D) 1:2
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The ratio of the number of freshmen to the number of seniors who attended the basketball game is 7 : 8.
What is the ratio?Ratio is used to show the relationship between two or more numbers. Ratio provides information on the frequency of one value within other values. The sign that is used to represent ratio is :.
The ratio of freshmen to juniors is 3 : 4.
The ratio of juniors to seniors is 7 : 6.
In order to determine the required values, let us make some assumptions.
The number of freshmen is 21
The number of juniors is 28.
Given the two above assumption, the number of seniors = (28 x 6) / 7 = 24
The ratio of freshmen to seniors = number of freshmen : number of seniors
21 : 24
Express the ratio in its simplest form - 7 : 8.
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z2=46Use a calculator.Z≈(Round to the nearest tenth as needed. Use a comma to separate answers as needed.)
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The given expression is
[tex]z^2=46[/tex]We can look for a number in which square power is 46 or close to 46.
Notice that, 6 times 6 is 36, and 7 times 7 is 49. This means The number is between 6 and 7.
[tex]z\approx6,-6[/tex]Now, we use a calculator to find the exact solution.
[tex]z=\sqrt[]{46}\approx\pm6.78[/tex]As you can see, the approximated solutions are 6.78 and -6.78.I need help on this calculus practice problem, I’m having trouble on it.
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From the question
We are given
[tex]\lim _{x\to-7}g(x)[/tex]We are to determine if the table below is appropriate for approximating the limit
From the table
The value of the limit as x tends to -7
Can be found using
[tex]x=-7.001\text{ and x = 7.001}[/tex]Hence, from the values given in the table
The table is appropriate
Find two consecutive whole numbers that 11 lies between.
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The number 11 is a whole number that lies between the whole numbers 10 and 12.
Therefore, the two consecutive whole numbers that 11 lies between are 10 and 12
Plot the complex number, then write the complex number in polar form. You may express the argument in degrees.
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DEFINITIONS
To represent a complex number we need to address the two components of the number.
Consider the complex number:
[tex]a+bi[/tex]Complex numbers are the points on the plane, expressed as ordered pairs (a, b), where a represents the coordinate for the horizontal axis and b represents the coordinate for the vertical axis.
Note that the imaginary part is plotted out on the vertical axis while the real part is on the horizontal axis.
QUESTION
The complex number is given to be:
[tex]4\sqrt[]{3}-4i[/tex]This means that the ordered pair representing the complex number is given to be:
[tex](a,b)=(4\sqrt[]{3},-4)[/tex]This means that the point will be positive on the real axis and negative on the imaginary axis. Therefore, the point will be in the 4th quadrant.
The correct option is OPTION B.
Prof. Glatt likes 2% milk (2% fat) for her cereal in the morning. Her parents only buy wholemilk (3.5% fat) and non-fat milk (0% fat). While she is visiting her parents, how much of eachtype of milk does she need to mix to get 3 cups of 2% milk. The answer can be rounded to thenearest tenth.linear systems solving algebraically
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It is given that there are two types of milk.
One is 3.5% and one is 0%.
Let the number of cups of 3.5% milk be x and the number of cups of 0% milk used be y.
The total should be 3 cups so it follows:
[tex]x+y=3\ldots(i)[/tex]It is also known that the resulting milk is 2% so it follows:
[tex]\begin{gathered} \frac{3.5}{100}x+\frac{0}{100}y=\frac{2}{100}(x+y) \\ \frac{3.5}{100}x=\frac{2}{100}(x+y) \end{gathered}[/tex]Multiply by 100 on both sides to get:
[tex]\begin{gathered} 3.5x=2(x+y) \\ 3.5x=2x+2y \\ 1.5x=2y \\ x=\frac{2}{1.5}y \\ x=\frac{2\times2}{1.5\times2}y \\ x=\frac{4}{3}y \end{gathered}[/tex]Substitute the value of (ii) in (i) to get:
[tex]\begin{gathered} x+y=3 \\ \frac{4}{3}y+y=3 \\ \frac{4+3}{3}y=3 \\ \frac{7}{3}y=3 \\ \frac{3}{7}\times\frac{7}{3}y=\frac{3}{7}\times3 \\ y=\frac{9}{7} \end{gathered}[/tex]Hence the quantity of 0% milk is 9/7 cups.
The quantity of 3.5% milk is given by:
[tex]\begin{gathered} x=\frac{4}{3}y \\ x=\frac{4}{3}\times\frac{9}{7} \\ x=\frac{12}{7} \end{gathered}[/tex]Hence the quantity of 3.5% milk is 12/7 cups.
Can you help me, please? I have to find the restrictions on x .Thank you.
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If angle A is greater than angle C, then side BC is greater than side AB.
[tex]\begin{gathered} BC>AB \\ 8-x>4x+48 \end{gathered}[/tex]Then, we solve for x
[tex]\begin{gathered} -4x-x>48-8 \\ -5x>40 \\ x<-\frac{40}{5} \\ x<-8 \end{gathered}[/tex]Hence, the restriction of x is that its value must be less than -8.hello I need help answering this homework question please thank you
Answers
Solution:
Case: Area
Given: A house to be painted
Method/ Final answers
a) Find the area of the garage to be painted.
(i) Front.
A = l X w - Garage door area
l= 15 ft, b= 10-4 gives 6ft
A= 15 X 6 - (10 X 7)
A= 90 - 70
A= 20 square feet
ii) Side
A= l X w
A= 6 X 5
A= 30 square feet
iii) The sum of areas
A= 20 + 30
A= 50 square feet
b) Area of the painted region around windows 5 and 6.
Since 12 in = 1 ft
Area of front door converted to feet is (20/3) ft by 3 ft
Areas of windows 5 and 6 converted to feet is 3 ft by (5/3) ft each
A= Area of space - (Area of front door + window 5 + window 6)
A= (30 X 10) - [(20/3) X 3 + 3 X (5/3) + 3 X (5/3)]
A= 300 - [20 + 5 + 5]
A= 300 - 30
A= 270 square feet.
c) Area of the painted region around windows 3
A= Total face - Area of window 3
A= (Rectangle + Parallelogram + Triangle) - Area of window 3
A= [(10 X 4) + (5 X 4) + (0.5 X 4 X 3)] - [1 X (5/3)]
A= [40+20+6] - [5/3]
A= 66 - (5/3)
A= 193/3 square feet
A= 63.33 square feet
d) Area of the region on the second floor with 2 rectangles and the region around window 4
i) region with rectangle 1 from left to right
A= 10 X (15- 6)
A= 10 X 9
A = 90
ii) region with rectangle 2 from left to right
A= 10 X (15- 6)
A= 10 X 9
A = 90
iii) Area of region around window 4
Area of space - area of window
A= 10 X (30-9-12) - [3 X (5/2)]
A= 10 X 9 - (15/2)
A= 82.5.
Total area= 90 + 90 + 82.5
= 262.5 square feet
e) Total area of the painted region (white)
262.5 + 63.33+ 270 + 50
= 645.83 square feet.
f) Additional question
The total cost if it cost $8 per sq ft
645.83 square feet X $8 per sq ft
=$5166.64
I inserted a picture of the question Check all that apply
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Recall that the line equation is of the form
[tex]y=mx+c\ldots\ldots\text{.}(1)[/tex]The points lie in the line are (2,5) and (-2,-5).
Setting x=2 and y=5 in the equa
A committee must be formed with 5 teachers and 4 students. If there are 6 teachers to choose from, and 15 students, how many different ways could the committee be made?
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There are 6 teachers and 15 students to choose from
To form a committee of 5 teachers and 4 students
The combination rule will be applied
From 6 teachers, The number of ways 5 teachers can be selected is
[tex]^6C_5[/tex]From 15 students, the number of ways 4 students can be selected is
[tex]^{15}C_{4^{}}[/tex]Therefore, the total number of ways a committee of 5 teachers and 4 students can be formed from 6 teachers and 15 students is
[tex]^6C_5\times^{15}C_{4^{}}[/tex]Simplifying this gives
[tex]^6C_5\times^{15}C_{4^{}}=\frac{6!}{(6-5)!\times5!}\times\frac{15!}{(15-4)!\times4!}[/tex]This further gives
[tex]\begin{gathered} ^6C_5\times^{15}C_{4^{}}=\frac{6!}{1!\times5!}\times\frac{15!}{11!\times4!} \\ ^6C_5\times^{15}C_{4^{}}=\frac{6\times5!}{1\times5!}\times\frac{15\times14\times13\times12\times11!}{11!\times4\times3\times2\times1} \end{gathered}[/tex]Cancel out common factors
[tex]\begin{gathered} ^6C_5\times^{15}C_{4^{}}=6\times\frac{15\times14\times13\times12}{4\times3\times2\times1} \\ ^6C_5\times^{15}C_{4^{}}=6\times\frac{32760}{24} \\ ^6C_5\times^{15}C_{4^{}}=6\times1365 \\ ^6C_5\times^{15}C_{4^{}}=8190 \end{gathered}[/tex]Therefore, the number of ways the committee can be formed is 8190 ways
Samantha drinks 2/3 gallon of water everyday.At this rate , how much water does Samantha drink in 30 days ?
Answers
Explanation:
We are told that Samantha drinks 2/3 gallon of water daily
We are then asked to find the volume of water that she will drink in 30 days
To do so, we will have:
[tex]\begin{gathered} if\text{ } \\ \frac{2}{3}gallons\text{ =1 day} \\ for\text{ } \\ 30\text{ days, this will be} \\ \frac{2}{3}gallons\text{ per day}\times30days\text{ =}\frac{60}{3}=20gallons \end{gathered}[/tex]Therefore, in 30 days, she will drink 20 gallons of water
The answer is 20 gallonsans
for each triangle list the sides in order from shortest to longest explain your reasoning with words or numbers for the order
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a. For the first triangle ,
Two angles are given. To determine the third angle apply the property of triangle which is sum of the angles of the triangle is 180 degree.
[tex]27^{\circ}+82^{\circ}+\angle T=180^{\circ}[/tex][tex]\angle T=180^{\circ}-27^{\circ}-82^{\circ}=71^{\circ}[/tex]The triangle angles and sides relationship-
The longest side of a triangle is opposite the biggest angle measure.
The shortest side of a triangle is opposite the smallest angle measure in a triangle.
Therefore,
The shortest side is side opposite to the smallest angle that is MT.
The largest side is side opposite to the largest angle that is AT.
Hence the order for the sides from shortest to largest is
[tex]MTb. The triangle is given . First determine the value of x and find the angle using the property of triangle which is sum of the angles of the triangle is 180 degree.
[tex]8x-1+3x+4+3x+9=180^{\circ}[/tex][tex]14x+12=180[/tex][tex]14x=168[/tex][tex]x=12[/tex]The angles obtained are
[tex](8x-1)=95,(3x+4)=40,(3x+9)=45[/tex]We know that the longest side of a triangle is opposite the biggest angle measure.
The shortest side of a triangle is opposite the smallest angle measure in a triangle.
Hence the shortest side is JK and largest side is JL.
Hence the order for the sides from shortest to largest is
[tex]JKThe function C(x) =17.5x-10 represents the cost (in dollars) of buying x tickets to the orchestra with a $10 coupon.
Answers
a) It represents the discount of $10 coupon
b) It repesetns the cost of each ticket
c) Five tickets cost is:
C(5) = 17.5(5) - 10 = 87.5 - 10 = 77.5
Five tickets cost $77.50
d) 130 = 17.5x - 10
130 + 10 = 17.5x
140 = 17.5x
x = 140/17.5 =8
x = 8
With $130 you can buy 8 tickets
Use Pythagorean theorem to find right triangle side lengthsFind the value of c in the triangle shown below.682Choose 1 answer:A = 28B= 64=9= 10
Answers
EXPLANATION
Given the Right Triangle, we can apply the Pythagorean Theorem in order to get the value of x as shown as follows:
[tex]\text{Hypotenuse}^2=Short_-leg^2+Long_-leg^2[/tex]Replacing terms:
[tex]x^2=6^2+8^2[/tex][tex]x^2=36+64=100[/tex]Applying the square root to both sides:
[tex]x=\sqrt[]{100}=10[/tex]Hence, the solution is x=10
4. Use the graph below to answer the following questions.
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By the concept of set builder notification
(b) 4
(c) 6
(d) -6
What is the set builder notification?In mathematics, the set-builder notation is used to express a set's elements or specify the conditions that each member of the set must meet. for instance The notation for the set builder is A = x Z | x 4 for the supplied set A =..., -3, -2, -1, 0, 1, 2, 3, 4. Sets can also be expressed using an interval or an equation by using the set-builder notation. In many cases, sets with an infinite number of elements have their elements written and represented in this way. These include integers, real numbers, and natural numbers, among other popular forms of numbers. One (or more) variables from these categories are utilized in this.
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You deposit $5000 in an account earning 6% interest compounded continuously. How much will you have in the account in 5 years?
Answers
For us to determine how much the account will be in 5 years at compounded continuously, we will be using the following formula:
[tex]\text{ A = P}_0e^{rt}[/tex]Where,
P = Principal amount (Initial Value)
A = Final amount (Future Value)
r = interest rate (in decimal)
t = time (in years)
e = mathematical constant approximately 2.7183
Given:
P = $5,000
r = 6% = 6/100 = 0.06
t = 5 years
We get,
[tex]\text{ A = P}_0e^{rt}[/tex][tex]\text{ A = (5,000)(2.7183)}^{(0.06)(5)}[/tex][tex]\text{ A = (5,000)(2.7183)}^{0.3}[/tex][tex]\text{ A = (5,000)(}1.34986151469)[/tex][tex]\text{ A = }6,749.30757343\text{ }\approx\text{ \$6,749.30}[/tex]Therefore, in 5 years, at 6% compounded continuously, your account will be $6,749.30
The following are the annual salaries of 15 chief executive offers of major companies. The salaries are written in thousands of dollars.
Answers
The original data is:
405, 1108, 84, 315, 495, 609, 362, 428, 224, 338, 700, 790, 814, 767, 633
To find the required percentiles, we need to sort the dataset from lowest to highest.
84, 224, 315, 338, 362, 405, 428, 495, 609, 633, 700, 767, 790, 814, 1108
The total number of data is 15.
a) The 25th percentile is the element located at the position:
25/100 * 15 = 3.75
Rounding down, the position is 3, so the 25th perc
I’m doing this timed assignment and I have no idea how to do it..help
Answers
Step 1
State the MACRS formula
[tex]Original\text{ value\lparen applicable MACRS \% rate\rparen}[/tex]Step 2
The original value=$20000
The depreciation for the first years is shown thus