Answer:
Given equation: [tex]y=x^2+1[/tex]
Therefore, we can say that any point on the curve has the coordinates [tex](a, a^2+1)[/tex] (where a is any constant)
To find the gradient of the tangent to the curve at any given point, differentiate the equation.
Given equation:
[tex]y=x^2+1[/tex]
[tex]\implies \dfrac{dy}{dx}=2x[/tex]
Therefore, the gradient at point [tex](a, a^2+1)[/tex] is [tex]2a[/tex]
Using the point-slope form of linear equation, we can create a general equation of the tangent at point [tex](a, a^2+1)[/tex]:
[tex]\begin{aligned}y-y_1 & =m(x-x_1)\\ \implies y-(a^2+1)& =2a(x-a)\end{aligned}[/tex]
[tex]\implies y=2ax-2a^2+a^2+1[/tex]
[tex]\implies y=2ax-a^2+1[/tex]
Given that the tangents pass through point (-1, -3), input this into the general equation of the tangent:
[tex]\begin{aligned}y &=2ax-a^2+1\\ \implies -3 & =2a(-1)-a^2+1\end{aligned}[/tex]
[tex]\implies 0=-2a-a^2+1+3[/tex]
[tex]\implies a^2+2a-4=0[/tex]
Use the quadratic formula to solve for a:
[tex]\implies a=\dfrac{-2\pm\sqrt{2^2-4(1)(-4)}}{2(1)}[/tex]
[tex]\implies a=\dfrac{-2\pm2\sqrt{5}}{2}[/tex]
[tex]\implies a=-1 \pm \sqrt{5}[/tex]
Input the found values of a into the general equation of the tangent to create the equations of the two lines:
[tex]\begin{aligned}a=-1+\sqrt{5}\implies y & =2(-1+\sqrt{5})x-(-1+\sqrt{5})^2+1\\ y & =(-2+2\sqrt{5})x-(6-2\sqrt{5})+1\\ y & =(-2+2\sqrt{5})x+2\sqrt{5}-5 \end{aligned}[/tex]
[tex]\begin{aligned}a=-1-\sqrt{5}\implies y & =2(-1-\sqrt{5})x-(-1-\sqrt{5})^2+1\\ y & =(-2-2\sqrt{5})x-(6+2\sqrt{5})+1\\ y & =(-2-2\sqrt{5})x-2\sqrt{5}-5 \end{aligned}[/tex]
Therefore, the equations of the two lines that pass through the point (-1, -3) and are tangent to the graph of [tex]y=x^2+1[/tex] are:
[tex]y=(-2+2\sqrt{5})x+2\sqrt{5}-5[/tex]
[tex]y=(-2-2\sqrt{5})x-2\sqrt{5}-5[/tex]
Solve the following equation
Answer:
Write it down in comments i can barely see
Step-by-step explanation:
A coin tossed four times and a card is selected from a standard deck of cards
Probability of landing on tails = 1/2Probability of choosing a spade = 1/4, as there are 13 spades in a pack of 52 cardsSo the probability of both = 1/2 * 1/4 = 1/8.We are assuming the deck of cards does not contain jokers.
Question in picture below
Answer:
f(x) = 3sin(3.1416(x +4)/7)
Step-by-step explanation:
You want the equation of a sine function with an amplitude of 3 and having one cycle between the points (-11, 0) and (3, 0), where it has negative slope.
EquationThe equation will generally be of the form ...
f(x) = (amplitude)·sin(2π/(period)·(x-(horizontal translation)))
ApplicationHere, the amplitude is 3 units, the period is (3 -(-11)) = 14 units, and the horizontal translation from the origin to the positive-going midline crossing is 4 units to the left.
Then f(x) is ...
f(x) = 3·sin(2π/14(x -(-4))
f(x) = 3·sin(3.1416/7(x +4))
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The regular price of a sofa, in dollars, is represented by . The sale price of the sofa is 30%off the regular price. Select alltrue statements.
A. The sale price of the sofa can be represented by −0.3.
B. The sale price of the sofa can be represented by 0.7.
C. The sale price of the sofa can be represented by 0.3
D. The dollar amount of the discount for the sale price can be represented by 0.3.
E. The dollar amount of the discount for the sale price can be represented by −0.3.
Answer:
BEStep-by-step explanation:
The sale price of the sofa is given as :
Original Price (100% - 30%)Original Price (70%)Original Price (0.7)The dollar amount for the discount
- 30% [discount]-0.3For two functions, f(x) and g(x), a statement is made that f(x) = g(x) at x = 5. What is definitely true about x = 5?
A. Both f(x) and g(x) cross the x-axis at 5.
B. Both f(x) and g(x) cross the y-axis at 5.
C. Both f(x) and g(x) have a maximum or minimum value at x = 5.
D. Both f(x) and g(x) have the same output value at x = 5
Answer:
D
Step-by-step explanation:
f(x) = g(x) at x=5
means
f(5) = g(5)
Please help I really need this
Answer:
(0 , 4)
Step-by-step explanation:
The Y-intercept is when a line crosses sector y at any point. This means x will be zero. Simply to find the y-intercept from this quadratic function, just look at where the line crosses the y-line.
This clearly passes at (0,4), therefore (0,4) is the y-intercept.
Pleaseeee answer I’ll give you pointsss
Answer:
Okay . That looks interesting. Can't help.
How would I solve I need to know Asap and what is the answer
Answer:
All of them are true.
Step-by-step explanation:
The symbol ≅ is used to show congruence between two shapes.
Congruency means that the shapes have the same shape and size.
This means that the two shapes are the same.
This means that all sides are going to be the same, so the first two statements are true.
This also means that the angles are the same, so the last two statements are also true.
mark brainliest please!
Answer:
1 and 3 are correct.
Step-by-step explanation:
You can just look at the order of the letters in the triangle congruence statement and see if they correspond with the answer choices.
For example... OC is congruent to JA because for both segments, it's the first letter, then the second letter from the beginning triangle congruence statement.
Which of the x values are solutions to the inequality 4(2 – x) > –2x – 3(4x + 1)? Check all that apply.
Step-by-step explanation:
[tex]4(2 - x) > - 2x - 3(4x + 1)[/tex]
[tex]8 - 4x > - 2x - 12x - 3[/tex]
[tex]8 - 4x > - 14x - 3[/tex]
[tex] - 4x + 14x > - 3 - 8[/tex]
[tex]10x > - 11[/tex]
[tex]x = - 1.1[/tex]
Answer:
[tex]x > \frac{-11}{10}[/tex]
Step-by-step explanation:
Step 1: Distribute
[tex]4(2 - x) > -2x - 3(4x + 1)[/tex]
[tex](4*2)+(4 * (-x)) > -2x + (-3 * 4x) + (-3 * 1)[/tex]
[tex]8 - 4x > -2x - 12x - 3[/tex]
Step 2: Combine like terms
[tex]8 - 4x > (-2x - 12x) - 3[/tex]
[tex]8 - 4x > -14x - 3[/tex]
Step 3: Add 14x to both sides
[tex]8 - 4x + 14x > -14x - 3 + 14x[/tex]
[tex]8 + 10x > -3[/tex]
Step 4: Subtract 8 from both sides
[tex]10x + 8 - 8 > -3 - 8[/tex]
[tex]10x > -11[/tex]
Step 5: Divide both sides by 10
[tex]\frac{10x}{10} > \frac{-11}{10}[/tex]
[tex]x > \frac{-11}{10}[/tex]
Answer: [tex]x > \frac{-11}{10}[/tex]
2x +2y-16x+4y+20=0 solve by completing the square
Answer
7/3
Step-by-step explanation:
Find f of x and f of y
F(x,y) = 2x + 27 - 16x + 4y + 20, f of x = -14
F(x,y) = 2x + 2y - 16x + 4y + 20, f of y = 6
Find the fist partial derivative
f of x = -14
f of y = 6
Subsitude f of x = 14 and f of y = 6 into dy/dx = - f of x / f of y
reduce the fraction
dy/dx = - 14/6 = - - 7/3 = 7/3
whats 1 added up to 1?
Mean:median:mode:range
Answer: Hi there ^^ oki so ima explain what these are/mean
So for starters this is used in math and isnt really useful but u got to do the work sadly so.
the mean is the arithmetic average of a set of given numbers.
the median is the middle in sorted numbers (after u sort the given numbers), ascending or desending, list of numbers can be more descripited of that data set than the average.
the mode is the value that appears most frequently, for example "the mode of {4,2,4,3,2,2} is 2 because it accurs 3 times, so in short words the mode is the number used the Most,
And lastly the Range is the differance between the highest and the lowest values in a set of numbers.
i hope this helps u out! ^^ ~xlXCherryColaXlx
Mrs. Murray wants to hang lights around the edge of the ceiling. The ceiling measures 14.5 feet by 9.6 feet. What is the perimeter, in feet, of Mrs. Murray's ceiling?
What is the perimeter of the ceiling in inches.
Answer:
48.2 feet
or
578.4 inches
Step-by-step explanation:
Perimeter is the distance around the shape or in this case the room. Assuming the room is a rectangle you would have two walls of the same length totaling 4 walls.
To find the perimeter add up 14.5 + 14.5 + 9.6 + 9.6 = 48.2 feet
Since it askes for the answer in inches you have to convert from feet to inches by multiplying your answer by twelve.
To make a small vase, Elisa uses no more than 4.5 ounces of clay. To make a large vase, she uses at least 12 ounces of clay. Which compound inequality represents the number of ounces of clay, c, that Elisa uses to make one vase of either size?
4.5 < c < 12
4.5 ≤ c ≤ 12
c < 4.5 or c > 12
c ≤ 4.5 or c ≥ 12
Answer:
last one is the answer of the following question
Answer:
D
Step-by-step explanation:
coulddd i be helped really quick?
So
For every increase in cover price of x means 5x+0.5 decreases customers by 30
So
the expression is
5x+0.5=300y-305x=300y-30.5300y=5x-30.55x-300y-30.5=0SOLVE PLSEEE I GIVE POINTS
Since the diameter for a segment is 68 centimeters long, we must divide that number by 4, since there are four circles. Next, we get 17.
Then, we use the formula for solving a circle, which is [tex]A = \pi r^2[/tex]. Plug in 8.5 (half of 17) for the radius, and then convert pi into 3.14. The equation should now look something like this: [tex]A = 3.14 * 8.5^2[/tex]. When solving, we get 265.865 for one circle only. We next multiply 256.856 by 4 since there are four circles. After multiplying you should get a total of 907.46 centimeters squared.
If this isn't correct, then do the same thing, but the radius is 34 centimeters.
Can any math experts please help me out with this? Thank you,
Answer:
13) x = 19
14) x = 12.4
Step-by-step explanation:
So, for 13, we know that the whole thing is 90 degrees, a right angle.
So, (4x - 17) + 31 = 90
Now, we'll subtract 31 from 90 to figure out the other angle.
4x - 17 = 59
Now, add seventeen to both sides to get 4x alone...
4x = 76
Divide each by four!
x = 19
Now, to check, we plug this x value into our first equation:
(4(19) - 17) + 31 = 90
(76 - 17) + 31 = 90
(59) + 31 = 90
90 = 90 :: true!
________________________________________
For 14, we know that the total must be 180, because it's a straight angle.
So, (5x + 22) + 96 = 180
Again, subtract the known angle from each side:
5x + 22 = 84
Subtract 22 from each side to get the exponent by itself...
5x = 62
Divide each side by 5:
x = 12.4
Now, plug it in to check.....
(5*12.4) + 22 + 96 = 180
62 + 22 + 96 = 180
84 + 96 = 180
180 = 180 :: true!
8 ≤ 2m Find variable of m
Please show your work
Answer:
m ≥ 4
Step-by-step explanation:
8 ≤ 2m
8/2 ≤ m
4 ≤ m
m ≥ 4
Describe a set of transformations that would map Triangle FGH onto Triangle F’G’H’
please help !! I dont know what the awnser is, i put mean, medium, and mode for the first try adn it gave me partial
Answer:
All of them but the last one
Step-by-step explanation:
find the derivative pls !!
Trey plans to watch 2 movies each month. Write an equation to represent the total number of movies n that he will watch in m months.
Answer:
2n*m. when there is a one in front of a letter, it is immediately just the letter
find the area for the shaded region
Answer:
610m^2 because that's math
Step-by-step explanation:
42x25 - 20x22
Step-by-step explanation:
S SOLUTION:
THE VOLUME OF CUBIOD IS 25 X 22 X 42=23100. M 3
VOLUME OF CUBIOD ; LENGTH X WIDTH X HEIGHT.
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this answer is correct
brain less this answer
Molly's alarm clock fell to the floor and the glass face was broken. Molly measured the glass face so that she could order a new one.
The radius of the circle-shaped glass face is 8 centimeters. What is the approximate area of the glass face? (Use 3.14 for .)
Answer:
200.96 cm^2
Step-by-step explanation:
πr^2 is the formula for the area of a circle
π*8^2
64π
substitute pi with 3.14
64 * 3.14 = 200.96
200.96
mark brainliest please!
(a) Is the image
of the dilation a reduction or an enlargement of the original figure? Explain.
(b) What is the scale factor? Explain.
the table shows the fees david charges for yard work
how much does david charge for yard work that takes 4.5 hours
Darcy pays $37.95 for a 25.3-kilometer taxi ride.
How much does Darcy pay, in dollars, per kilometer?
Answer:
1.5 per kilo
Step-by-step explanation:
Answer: That would be $1.50
Explanation: I took a test and I got it right
What is the surface area of the cylinder with height 3 yd and radius 4 yd? Round your
answer to the nearest thousandth.
A≈175.93yd²
A=2πrh+2πr2=2·π·4·3+2·π·42≈175.92919yd²
Therfore the surface area of the cylinder with height 3 yd and radius 4 yd is 175.93
Simplify this expression . With each step. please (x2 - x - 2)/(x2 + 5x + 4)
Answer:
I hope dis helps
Step-by-step explanation:
Simplify
x2 + 5x + 4
Trying to factor by splitting the middle term
1.1 Factoring x2 - x - 2
The first term is, x2 its coefficient is 1 .
The middle term is, -x its coefficient is -1 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 1 • -2 = -2
Step-2 : Find two factors of -2 whose sum equals the coefficient of the middle term, which is -1 .
-2 + 1 = -1 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -2 and 1
x2 - 2x + 1x - 2
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-2)
Add up the last 2 terms, pulling out common factors :
1 • (x-2)
Step-5 : Add up the four terms of step 4 :
(x+1) • (x-2)
Which is the desired factorization
Trying to factor by splitting the middle term
1.2 Factoring x2+5x+4
The first term is, x2 its coefficient is 1 .
The middle term is, +5x its coefficient is 5 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 1 • 4 = 4
Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is 5 .
-4 + -1 = -5
-2 + -2 = -4
-1 + -4 = -5
1 + 4 = 5 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 4
x2 + 1x + 4x + 4
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+1)
Add up the last 2 terms, pulling out common factors :
4 • (x+1)
Step-5 : Add up the four terms of step 4 :
(x+4) • (x+1)
Which is the desired factorization
Canceling Out :
1.3 Cancel out (x+1) which appears on both sides of the fraction line.
Final result :
x - 2
—————
x + 4
What is the length of the hypotenuse of the triangle below?
45
h
45°
5√2
90⁰
5√2
Answer:
10
Step-by-step explanation:
find the square root of the sum of the squares
The length of the hypotenuse for the given right angled triangle is 10 units.
What is a Pythagoras theorem?Pythagoras theorem states that the square of the hypotenuse is equivalent to the square of the base and perpendicular.
We can write the formula for Pythagoras theorem as -
(hypotenuse)² = (base)² + (perpendicular)²
Given is to find the length of the hypotenuse in the right angled triangle shown.
Using the Pythagoras theorem, we can write that -
(hypotenuse)² = (base)² + (perpendicular)²
(hypotenuse)² = (5√2)² + (5√2)²
(hypotenuse)² = 2 x (5√2)²
(hypotenuse)² = 2 x 25 x 2
(hypotenuse)² = 100
(hypotenuse) = 10
Therefore, the length of the hypotenuse for the given right angled triangle is 10 units.
To solve more questions on triangles, visit the link-
https://brainly.com/question/2773823
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