The function h(x) = x2 + 14x + 41 represents a parabola. Part A: Rewrite the function in vertex form by completing the square. Show your work. (6 points) Part B: Determine the vertex and indicate whether it is a maximum or a minimum on the graph. How do you know? (2 points) Part C: Determine the axis of symmetry for h(x). (2 points)
Answers
Part A:
The first thing of completing the square is writing the
expression
as
which expands to
.
We have the first two terms exactly alike with the function
we start with:
and
but we need to add/subtract from the last term, 49, to
obtain 41.
So, the second step is to subtract -8 from the expression
The function in finalizing the square form is
Part B:
The vertex is acquired by equating the expression in the
bracket from part A to zero
It means the curve has a turning point at x = -7
This vertex is a minimum since the function will make a
U-shape.
A quadratic function
can either make U-shape or ∩-shape depends on the
value of the constant
that goes with
. When
is (+), the curve is U-shape. When
(-), the curve is ∩-shape
Part C:
The symmetry line of the curve will go through the vertex,
hence the symmetry line is
This function is shown in the diagram below
Answer:
Part A: The vertex form is h(x) = (x+7)^2 - 8 .
Part B: The vertex is a minimum. The vertex is (-7,-8).
Part C: The axis of symmetry is x=-7. (axis of symmetry is the x value of the vertex)
Step-by-step explanation:
This video will help you understand how I got the function into vertex form.
Search: How do you convert from standard form to vertex form of a quadratic Brian McLogan
This will help you find the vertex
Search: Finding the vertex of a parabola in standard form khan academy
Related Questions
Three cities lie along a perfectly linear route: Springfield, Clarksville, and Allentown. Molly lives in Springfield and works in Allentown. She makes it to work using two gallons of gas in her car. Her friend Edgar lives in Allentown and works in Clarksville. It takes Edgar one gallon of gas to get to work. If Molly's car averages 26 miles per gallon, and Edgar's car averages 17 miles per gallon, about how far apart are Springfield and Clarksville?
Answers
Now, since Molly travelled from Springfield to Allentown, the total distance she covered is the distance Edward travelled and the unknown distance between Springfield and Clarksville, denoted as x in the picture. So, the equation is then
Molly's distance = x + Edgar's distance
26 miles/gal(2 gal) = x + 17 miles/gal(1 gal)
x = 35 miles
The best approximation for the square root of 10 is.. A).5 B).100 C).3.1 D).25
Answers
Answer:
It is approximately 3.1
Step-by-step explanation:
Find PS if ABC=PQR, AD is an altitude of ABC, PS is an altitude of PQR, AD=12, AC=16 and PR=10
a. 7.5
b. 19.2
c. 4.62
d. 19.5
Answers
First you need to find how they're similar.
16/10 = 1.6
Then, multiply 12 (AD) by 1.6 to find PS.
12 * 1.6 = 19.2
The answer is B. 19.2
Barry’s Bagel Emporium sells a dozen bagels for $5.00. This price is no longer high enough to create a profit. The owner decides to raise the price. He does not want to alarm his customers with too large of an increase. He is considering four different plans. Plan A: Raise the price by $0.05 each week until the price reaches $8.00. Plan B: Raise the price by 10 percent each week until the price reaches $8.00. Plan C: Raise the price by the same amount each week for 6 weeks, so that in the sixth week the price is $8.00. Plan D: Raise the price by $0.25 each week until the price reaches $8.00. Which plan will result in the price of the bagels reaching $8.00 fastest? plan A plan B plan C
Answers
Hope This Helps
Correct If I'm Wrong
Answer:
Plan B is correct answer.
Step-by-step explanation:
Raise the price by 10 percent each week until the price reaches $8.00.
Week 1. Starting price $5
[tex]0.1\times5=0.5[/tex]
price becomes = [tex]5+0.5=5.5[/tex]
Week 2.
[tex]0.1\times5.5=0.55[/tex]
Price becomes = [tex]5.5+0.55=6.05[/tex]
Week 3.
[tex]0.1\times6.05=0.605[/tex]
Price become = [tex]6.05+0.605=6.655[/tex]
Week 4.
[tex]0.1\times6.655=0.6655[/tex]
Price becomes = [tex]6.655+0.6655=7.320[/tex]
Week 5.
[tex]0.1\times7.320=0.732[/tex]
Price becomes = [tex]7.320+0.732=8.052[/tex]
So, we can see that in 5 weeks the price becomes $8 from $5. Therefore, plan B is the best plan.
Conditional probabilities are based on some event occurring given that something else has already occurred?
Answers
The answer is true. A conditional probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has occurred. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A in the condition B", is usually written as P (A|B). The conditional probability of A given B is well-defined as the quotient of the probability of the joint of events A and B, and the probability of B.
Determine whether the equation represents y as a function of x 16x-y^4=0
Answers
[tex]16x-y^4=0 \\ \\ \Rightarrow y^4=16x \\ \\ \Rightarrow y= \sqrt[4]{16x} =2 \sqrt[4]{x} [/tex]
For any value of x, there are two possible values of y, thus the equation does not represent y as a function of x.
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Which of the following is a solution of x2 + 4x + 10?
2 + i times the square root of 6
−2 + i times the square root of 24
−2 + i times the square root of 6
2 + i times the square root of 24
Answers
x^2+4x=-10
x^2+4x+4=-6
(x+2)^2=-6
x+2=±i√6
x=-2±i√6
So the correct answer is the third one down from the top.
Answer:
[tex]x=2+-i \sqrt{6}[/tex]
Step-by-step explanation:
[tex]x^2 + 4x + 10[/tex]
To find out the solution we set the expression =0 and solve for x
[tex]x^2 + 4x + 10=0[/tex]
Apply quadratic formula to solve for x
[tex]x=\frac{-b+-\sqrt{b^2-4ac}}{2a}[/tex]
a=1, b=4, c=10 plug in the values in the formula
[tex]x=\frac{-4+-\sqrt{4^2-4(1)(10)}}{2a}[/tex]
[tex]x=\frac{-4+-\sqrt{-24}}{2(1)}[/tex]
The value of square root (-1) is 'i'
[tex]x=\frac{-4+-2i\sqrt{6}}{2}[/tex]
Divide each term by 2
[tex]x=2+-i\sqrt{6}[/tex]
Convert this percent into decimal form.
Answers
Then move the decimal place two places to the left to divide by 100 (percent means per 100),
Final answer: 0.1225
75% of our 1000 products are shipped on time each month the remainder have defects that take two weeks to fix and ship our clients complain about 10% of the anti-products are defective and 5% of the product shipped late or defective what is the overall percentage of defective products
Answers
The number of products shipped late is 25% of 1000 = 250
The number of defective product from the products that shipped on-time is 10% of 750 = 75
The number of defective product from the products that shipped late is 5% of 250 = 12.5
The total number of defective products is 75+12.5 = 87.5.
This is 87.5 out of 1000 products and as percentage it's [tex] \frac{87.5}{1000}=0.0875 [/tex]×100 = 8.75%
Final answer:
Calculating the overall percentage of defective products from the given data, we find that 150 out of 1000 products are defective, leading to an overall defect rate of 15%.
Explanation:
The question asks us to calculate the overall percentage of defective products based on the given scenarios. Firstly, it's mentioned that 75% of 1000 products are shipped on time, which means 750 are shipped on time and 250 are initially defective.
Since clients complain that 10% of the products are defective and 5% of the products are shipped late or are defective, we need to consider these percentages in our calculations.
To find the number of defective products, we can assume the 10% complaint rate on the entire batch of products which would lead to 100 out of 1000 products being defective. This is the initial estimated number of defective products.
To address the 5% of the products that are both shipped late and are defective, we consider this as an additional defect rate on top of the existing one, which would be another 50 products.
Total defects would then be the sum of defects from the complaints about defects and the defects because of shipping delay, which amounts to 100 + 50 = 150 defective products. To find the overall percentage, we divide 150 by 1000 and multiply by 100, giving us an overall defect rate of 15%.
The length of a rectangle is 2 yd longer than its width. if the perimeter of the rectangle is 40 yd , find its area.
Answers
perimeter = 2L+2W
L=2+w
40 = 2L+2W
40= 2(2+w)+2W
40=4+2w+2w
36=4w
w=9
L=9+2=11
2(9) = 18, 2(11) = 22, 22+18 = 40
L=11
W=9
Area = L x w
area = 11x9= 99 square yards
Find the base of a parallelogram with an area (
a. of 60 square inches and height (h) of 4 inches. Use the formula for the area of a parallelogram
Answers
base = area of parallelogram/ height
= 60/4
= 15 inches
The base of parallelogram is 15 inches.
What is parallelogram?A parallelogram is a two-pair quadrilateral with parallel sides. A parallelogram has opposite sides that are the same length and have opposite angles that are the same size. Additionally, the interior angles on the same transversal side are supplemental. 360 degrees is the sum of all the interior angles.
Given area of a parallelogram is 60 square inches
area of parallelogram is given by product of base and height,
Area = b × h
where b = base and h = height
height = 4 inches
Area = b × h
60 = b × 4
b = 60/4 = 15 inches
Hence the base is 15 inches.
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Assume that y varies inversely with x. If y=7 when x=2/3, find y when x=7/3
Answers
y=k/x, which we can express as:
yx=k we are given the point (2/3, 7) so we can solve for k
7(2/3)=k
14/3=k
y=14/(3x), so when x=7/3
y=(14/3)/(7/3)
y=(14/3)(3/7)
y=2
[tex] \frac{x}{5}+\frac{3x}{15}=\frac{2x}{3} } [/tex]+2 Answer plz math help
Answers
3x+3x= 10x+30
Combine like terms
6x=10x+30
Subtract 10x from both sides
-4x=30
Divide both sides by -4
x= -7.5
Final answer: -7.5
what is the inverse of the function f(x)=1/9x+2
Answers
y = 1/9x + 2,
you should swap the places of x and y and find y again, in this case it will be
x = 1/9y + 2 => x-2 = 1/9y => (x-2) * 9y = 1 => 9y = 1/(x-2)
and finally we got y = [tex] \frac{1}{(x-2)*9} [/tex]
Tell which equation you would use to isolate a variable in order to solve the system using substitution. Explain your reasoning.
2x + y=-10
3x-y=0
Answers
Multiply both side by 3
2x*3+y*3=-10*3
6x+3y=-30
3x-y=0
Multiply both side by 2
3x*2-y*2=0*2
6x-2y=0
Next, use substitute property
6x+3y=-30
-
6x-2y=0
=
y=-30
3x-y=0
Substitute y with -30
3x-30=0
Add 30 to each side
3x-30+30=0+30
3x=30
Divided both side by 3
3x/3=30/3
x=10, so the solution pair is (10,-30). In this case, there is the first way to solve these two equation.
Then, I would use equation 3x+1.5y=-15 in this question by multiply 2x+y=-10 by multiplying both side by 3/2 to eliminate x and to solve variables for y. Hope it help!
If a fair coin is tossed 9 times, in how many different ways can the sequence of heads and tails appear
Answers
10 different sequences
all tails, 1 head 8 tails,2 heads 7 tails,3 heads 6 tails,4 heads 5 tails,5 heads 4 tails,6 heads 3 tails,7 heads 2 tails,8 heads 1 tail and All heads.
HONORS PROJECT GEOMETRY HELP
Maurice and Johanna have appreciated the help you have provided them and their company Pythgo-grass. They have decided to let you consult on a big project.
1, A triangular section of a lawn will be converted to river rock instead of grass. Maurice insists that the only way to find a missing side length is to use the Law of Cosines. Johanna exclaims that only the Law of Sines will be useful. Describe a scenario where Maurice is correct, a scenario where Johanna is correct, and a scenario where both laws are able to be used. Use complete sentences and example measurements when necessary.
2, An archway will be constructed over a walkway. A piece of wood will need to be curved to match a parabola. Explain to Maurice how to find the equation of the parabola given the focal point and the directrix.
3, There are two fruit trees located at (3,0) and (−3, 0) in the backyard plan. Maurice wants to use these two fruit trees as the focal points for an elliptical flowerbed. Johanna wants to use these two fruit trees as the focal points for some hyperbolic flowerbeds. Create the location of two vertices on the y-axis. Show your work creating the equations for both the horizontal ellipse and the horizontal hyperbola. Include the graph of both equations and the focal points on the same coordinate plane.
4, A pipe needs to run from a water main, tangent to a circular fish pond. On a coordinate plane, construct the circular fishpond, the point to represent the location of the water main connection, and all other pieces needed to construct the tangent pipe. Submit your graph. You may do this by hand, using a compass and straight edge, or by using a graphing software program.
5, Two pillars have been delivered for the support of a shade structure in the backyard. They are both ten feet tall and the cross sections of each pillar have the same area. Explain how you know these pillars have the same volume without knowing whether the pillars are the same shape.
Answers
The expression 9n is also considered a _____.
constant
variable
term
Answers
Answer:
Term
Step-by-step explanation:
hope this helps
Given the arithmetic sequence A1, A2, A3, A4, 58,69,80,91
What is the Value of A21?
***PLEASE HELP******
Answers
i.e. a, a+d, a+2d, a+3d...
where a is the first term, and d is the difference
in your case, a (first term) = 58
to find the difference, subtract 58 (a) from 69 (a+d) to get d.
d = 69 - 58 = 11
(you can always check your calculation on another two consecutive terms: 91 - 80 also equals 11)
the general formula for any term in an arithmetic sequence is:
T(n) = a + (n-1)d ...(or A(n))
in your case, n = 21
subbing into the formula:
A21 = a + (21-1)d = 58 + 20(11) = 278
hope that helps :)
Given f(x) = x2 + 4x − 1 and g(x) = 5x − 7, identify (fg)(x).
Answers
(fg)(x) = (x^2 + 4x − 1)(5x − 7)
(fg)(x) = 5x^3 + 20x^2 - 5x - 7x^2 - 28x + 7
(fg)(x) = 5x^3 + 13x^2 - 33x + 7
answer is C. third choice
(fg)(x) = 5x^3 + 13x^2 - 33x + 7
The product of the functions[tex]\( f(x) = x^2 + 4x - 1 \) and \( g(x) = 5x - 7 \) is \( 5x^3 + 13x^2 - 33x + 7 \).[/tex]
The correct answer is indeed [tex]{C} \),[/tex] which matches [tex]\( 5x^3 + 13x^2 - 33x + 7 \).[/tex]
To find the product[tex]\( (f \cdot g)(x) \)[/tex], where [tex]\( f(x) = x^2 + 4x - 1 \)[/tex] and [tex]\( g(x) = 5x - 7 \),[/tex]we need to perform the multiplication of these two functions.
Start by expanding [tex]\( f(x) \cdot g(x) \):[/tex]
1. Write down ( f(x) ):
[tex]\[ f(x) = x^2 + 4x - 1 \][/tex]
2. Write down ( g(x) ):
[tex]\[ g(x) = 5x - 7 \][/tex]
3. Perform the multiplication [tex]\( f(x) \cdot g(x) \)[/tex]:
[tex]\[ f(x) \cdot g(x) = (x^2 + 4x - 1)(5x - 7) \][/tex]
4. Distribute [tex]\( x^2 + 4x - 1 \)[/tex] across ( 5x - 7 ):
[tex]\[ f(x) \cdot g(x) = x^2 \cdot (5x - 7) + 4x \cdot (5x - 7) - 1 \cdot (5x - 7) \][/tex]
5. Perform the multiplications:
[tex]\[ x^2 \cdot (5x - 7) = 5x^3 - 7x^2 \][/tex]
[tex]\[ 4x \cdot (5x - 7) = 20x^2 - 28x \][/tex]
[tex]\[ -1 \cdot (5x - 7) = -5x + 7 \][/tex]
6. Combine all the terms:
[tex]\[ f(x) \cdot g(x) = 5x^3 - 7x^2 + 20x^2 - 28x - 5x + 7 \][/tex]
7. Simplify by combining like terms:
[tex]\[ f(x) \cdot g(x) = 5x^3 + (20x^2 - 7x^2) + (-28x - 5x) + 7 \][/tex]
[tex]\[ f(x) \cdot g(x) = 5x^3 + 13x^2 - 33x + 7 \][/tex]
Therefore, the product [tex]\( (f \cdot g)(x) \) is \( 5x^3 + 13x^2 - 33x + 7 \).[/tex]
The correct answer is indeed [tex]{C} \),[/tex] which matches [tex]\( 5x^3 + 13x^2 - 33x + 7 \).[/tex]
A drawer contains five pairs of socks that are brown, black, white, red, and blue. Claude takes the red socks out of the drawer. What is the probability of Claude choosing the red socks on his first pick?
Answers
A drawer contains five pairs of socks that are brown, black, white, red, and blue. Claude takes the red socks out of the drawer. What is the probability of Claude choosing the red socks on his first pick?
Answer: 1/25
Answer:
The answer would be 1/25 Hopefully this any T4L students!
last question
help me pls c:
Answers
Identify intervals on which the function is increasing, decreasing, or constant. g(x) = 4 - (x - 6)^2 ??
Answers
g(x)=4-(x-6)^2
g(x)=4-(x^2-12x+36)
g(x)=4-x^2+12x-36
g(x)=-x^2+12x-32
dg/dx=-2x+12
So g(x) will be increasing when dg/dx>0
-2x+12>0
-2x>-12
x<6
So g(x) is increasing on the interval (-oo, 6)
g(x) will be decreasing when dg/dx<0
-2x+12<0
-2x<-12
x>6
So g(x) will be decreasing on the interval (6, +oo)
The linear equation when b = 5 and m = –2 is
Answers
So the answer would be y=-2x+5 so the answer would be D
Answer:
y=-2x+5
Step-by-step explanation:
Which geometric series converges?
A. 2+0.2+0.02+0.002+...
B. 2+4+8+16+...
C. 2-20+200-2000+...
D. 2+2+2+2+...
Answers
Answer: The correct option is (A) 2 + 0.2 + 0.02 + 0.002 + . . .
Step-by-step explanation: We are given to select the correct geometric series that converges.
We know that
a geometric series converges if the modulus of its common ratio is less than 1.
Option (A) : 2 + 0.2 + 0.02 + 0.002 + . . .
Here, first term, a= 2 and the common ratio is given by
[tex]r=\dfrac{0.2}{2}=\dfrac{0.02}{0.2}=\dfrac{0.002}{0.02}=~.~.~.~=0.1\\\\\Rightarrow |r|=|0.1|=0.1<1[/tex]
So, this geometric series will converge.
Option (A) is correct.
Option (B) : 2 + 4 + 8 + 16 + . . .
Here, first term, a= 2 and the common ratio is given by
[tex]r=\dfrac{4}{2}=\dfrac{8}{4}=\dfrac{16}{8}=~.~.~.~=2\\\\\Rightarrow |r|=|2|=2>1.[/tex]
So, this geometric series will not converge.
Option (B) is incorrect.
Option (C) : 2 - 20 + 200 - 2000 + . . .
Here, first term, a= 2 and the common ratio is given by
[tex]r=\dfrac{-20}{2}=\dfrac{200}{-20}=\dfrac{-2000}{200}=~.~.~.~=-10\\\\\Rightarrow |r|=|-10|=10>1.[/tex]
So, this geometric series will not converge.
Option (C) is incorrect.
Option (D) : 2 +2 + 2 + 2 + . . .
Here, first term, a= 2 and the common ratio is given by
[tex]r=\dfrac{2}{2}=\dfrac{2}{2}=\dfrac{2}{2}=~.~.~.~=1\\\\\Rightarrow |r|=|1|=1.[/tex]
So, this geometric series will not converge.
Option (D) is incorrect.
Thus, the correct option is (A).
The geometric series that converges is 2+0.2+0.02+0.002+ ...:
Thus, option (A) is correct.
In a geometric series, the terms are multiplied by a constant ratio to obtain the next term.
If the absolute value of the common ratio is less than 1, the series converges.
A. 2 + 0.2 + 0.02 + 0.002 + ...:
In this series, the common ratio is 0.1 (each term is divided by 10).
Since the absolute value of the common ratio is less than 1, this series converges.
B. 2 + 4 + 8 + 16 + ...:
In this series, the common ratio is 2 (each term is multiplied by 2).
Since the common ratio is greater than 1, this series diverges.
C. 2 - 20 + 200 - 2000 + ...:
In this series, the terms alternate in sign, but the absolute value of the common ratio is 10.
Since the absolute value of the common ratio is greater than 1, this series diverges.
D. 2 + 2 + 2 + 2 + ...:
In this series, the common ratio is 1 (each term is the same).
Since the common ratio is equal to 1, this series neither converges nor diverges. It is a divergent series.
Therefore, the geometric series that converges is 2 + 0.2 + 0.02 + 0.002 + ...
Thus, option (A) is correct.
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P, Q, and R are three different points. PQ = 3x + 2, QR = x, RP = x + 2, and . List the angles of PQR in order from largest to smallest and justify your response.
Answers
QR = x ⇒ x>0
If x>0 then:
QR is the smallest side
PQ is the largest side
In the triangle, the largest angle lies opposite the largest side.
The angles of ΔPQR in order from largest to smallest:
∠R is largest [opposite to PQ]
∠Q is middle [opposite to RP]
∠P is smallest [opposite to QR]
The angles of triangle PQR are ordered from largest to smallest as <>R,
The question involves applying the principles of geometry to compare lengths of sides in a triangle, and thereby determine the relative magnitude of angles in triangle PQR. According to the triangle inequality theorem, the largest angle in a triangle is opposite the longest side, and the smallest angle is opposite the shortest side. Given that the side lengths are represented as PQ = 3x + 2, QR = x, and RP = x + 2, we can compare the expressions to conclude which side is longest and which is shortest, assuming all values of x are positive since they represent lengths.
Firstly, it is obvious that QR (x) is the shortest side since it is just x without any additional positive value. Secondly, between RP (x + 2) and PQ (3x + 2), PQ will always be longer than RP for all positive x because it has a larger coefficient in front of x. Hence, the angle opposite PQ (angle R) will be the largest angle, and the angle opposite QR (angle P) will be the smallest. The angle at Q will be between the other two angles in terms of their measurements since PQ is longer than RP but both are longer than QR.
To summarize, the angles of Triangle PQR ordered from largest to smallest are: ∠R, ∠Q, and ∠P.
Which of the following points lie in the solution set to the following system of inequalities?
y ≤ x − 5
y ≥ −x − 4
(−5, 2)
(5, −2)
(−5, −2)
(5, 2)
Answers
Answer: Second option : (5, −2)
Step-by-step explanation: Given system of inequalities
y ≤ x − 5
y ≥ −x − 4
Plugging x=5 and y=-2 in first inequality
-2 ≤ 5 − 5
-2 ≤ 0 : True.
Plugging x=5 and y=-2 in second inequality
-2 ≥ −5 − 4
-2 ≥ -9 : Also true.
Point (5, −2) satisfied both of the given inequalities in the system.
Therefore, (5,-2) is correct option.
Which function below is the inverse of f(x) = The quantity of four x minus three, over two.?
Answers
99 POINTS!!! Find the equation for an ellipse with vertices at (-6, 0) and (6, 0) and foci at (-4, 0) and (4, 0).
Answers
a>b
a=6, a^2=36
foci=(a^2-b^2)^(1/2)
4=(36-b^2)^(1/2)
16=36-b^2
b^2=36-16
b^2=20
b=2(5)^(1/2) or (20)^(1/2)
1=(x^2/36)+(y^2/20)
(x^2)/a^2+(y^2)/b^2=1
a>b
a=6, a^2=36
foci=(a^2-b^2)^(1/2)
4=(36-b^2)^(1/2)
16=36-b^2
b^2=36-16
b^2=20
b=2(5)^(1/2) or (20)^(1/2)
1=(x^2/36)+(y^2/20)
PLEASE HELP!!!!! The velocity of sound in air is given by the equation , where v is the velocity in meters per second and t is the temperature in degrees Celsius. Find the temperature when the velocity is 329 meters per second by graphing the equation. Round the answer to the nearest degree. Show your work.
Answers
The equation of velocity of sound in air is v = 20 √(273 + t). In this problem, we need to find the temperature when the velocity is 329 meters/s. You need to measure the time it takes a sound to travel a measured distance in order to measure its speed in air.
Given:
Velocity of sound in air equation = v = 20 √(273 + t)
Velocity = 329 m/s
To solve:
V = 20 √(273 + t)
329 = 20 * sqrt(273 + t)
16.45 = sqrt(273 + t)
273 + t = 16.45^2
t- 16.45^2-273
t = -2.4 degrees Celsius
So, the temperature when the velocity is 329 meters per second is -2.4 degrees Celsius.