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
[tex]h(x)= x^{2} +14x+41[/tex]
The first step of completing the square is writing the expression [tex] x^{2} +14x[/tex] as [tex] (x+7)^{2} [/tex] which expands to [tex] x^{2} +14x+49[/tex].
We have the first two terms exactly the same with the function we start with: [tex] x^{2} [/tex] and [tex]14x[/tex] 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 [tex] x^{2} +14x+49[/tex]
The function in completing the square form is
[tex]h(x)= (x+7)^{2}-8 [/tex]
Part B:
The vertex is obtained by equating the expression in the bracket from part A to zero
[tex]x+7=0[/tex]
[tex]x=-7[/tex]
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 [tex]a x^{2} +bx+c[/tex] can either make U-shape or ∩-shape depends on the value of the constant [tex]a[/tex] that goes with [tex] x^{2} [/tex]. When [tex]a[/tex] is (+), the curve is U-shape. When [tex]a[/tex] (-), the curve is ∩-shape
Part C:
The symmetry line of the curve will pass through the vertex, hence the symmetry line is [tex]x=-7[/tex]
This function is shown in the diagram below
to anyone who may be a lil' sus, the previous answer is correct.
Related Questions
Find the value of each variable for the right triangles. Make sure all answers are in reduced radical from. Shoe your work.
Answers
a.
[tex] t^{2}+ 12^{2}= 13^{2} [/tex]
instead of taking the square of 12 and 13, , take 12 squared to the right side and apply the difference of squares formula: [tex] a^{2}- b^{2}=(a-b)(a+b)[/tex]
[tex]t^{2}= 13^{2}- 12^{2}=(13-12)(13+12)=1*25[/tex]
so t is square root of 25, which is 5
b.
[tex] a^{2}+9 ^{2} = 12^{2} [/tex]
[tex]a^{2}= 12^{2} -9 ^{2}=(12-9)(12+9)=3*21=3*3*7 [/tex]
[tex]a= \sqrt{3*3*7}=3 \sqrt{7} [/tex]
b.
[tex] 6^{2} + 9^{2} = x^{2} [/tex]
[tex]36+81= x^{2} [/tex]
here we can factorize 9, to work with small numbers:
[tex]9(4+9)= x^{2}[/tex]
[tex]9*13= x^{2} [/tex]
[tex] x^{2} =3*3*13[/tex]
[tex]x= \sqrt{3*3*13}=3 \sqrt{13} [/tex]
monique deposited 1800 at the beginning of each quarter for 6 years into an ira. given an interest rate of 6% compounded quarterly find the value
Answers
The formula is
Fv=pmt [((1+r/k)^(kn)-1)÷(r/k)]×(1+r/k)
Fv future value?
PMT amount deposited 1800
R interest rate 0.06
K compounded quarterly 4
N time 6years
Fv=1,800×((((1+0.06÷4)^(4×6)
−1)÷(0.06÷4))×(1+0.06÷4))
=52,313.44....answer
A restaurant earns $1073 on Friday and $1108 on Saturday. Write and solve an equation to find the amount xx (in dollars) the restaurant needs to earn on Sunday to average $1000 per day over the three-day period. Write your equation so that the units on each side of the equation are dollars per day
Answers
(2181 + x) / 3 = 1000 ....multiply both sides by 3
2181 + x = 1000 * 3
2181 + x = 3000
x = 3000 - 2181
x = 819 <== they would need to earn $ 819 on Sunday
What is the discriminant of 3x2-10x=-2 ?
Answers
The discriminant of [tex]3x^2-10x=-2[/tex] is 76
To find the discriminant of [tex]3x^2-10x=-2[/tex], we have to recall the quadratic formula.
Further ExplanationThe formula of the quadratic equation can be expressed as [tex]ax^2+bx+c=0[/tex] and discriminant also expressed as [tex]D=b^2-4ac[/tex]
Using the quadratic formula, the given question can be expressed as:
[tex]3x^2-10x+2=0[/tex]
Where a = 3, b = -10, and c = 2
If we substitute the values, then the discriminant would be:
Recall the discriminant is [tex]D=b^2-4ac[/tex]
Therefore, we have
[tex]D=(-10)^2-4(3)(2)[/tex]
[tex]D=100-24[/tex]
[tex]D=76[/tex]
Thus, the discriminant of [tex]3x^2-10x=-2[/tex] is 76.
Discriminant is very helpful in quadratic equation as it helps reveals the number of real solutions to a given quadratic equation. In other words, it helps you to solve quadratic equation.
While discriminant only tells only reveals the number of real solutions, it doesn’t reveal what the solutions are. The determinant would only give you hint as regards the number of real solution to look for.
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discriminantquadratic formula3x2-10x=-2solutionquadratic equationtristen is a saleman whose base pay plus commissions amounted to $59,238 last year. if his base pay was $31,050 and he made $97,200 is sales last year what was his rate of commission?
Answers
subtract base pay from what he made
59238-31050 = 28188
so he made 28188 in commissions
now divide that by 97200
28188/97200 = 0.29
so he makes 29% commision
x plus 20% = 1980
what is x
Answers
x + 0.2x = 1980
1.2x = 1980
x = 1980/1.2
x = 1650 ← unknown number
What positive number is the same as its reciprocal?
Answers
The only positive number that is the same as its reciprocal is 1. This is achieved by setting up the equation x = 1/x and solving it.
Explanation:In mathematics, a number and its reciprocal are the same when the number is 1. If you denote the mysterious number we are looking for as 'x', then its reciprocal would be 1/x. Since the number and its reciprocal are the same, we can set up the equation x = 1/x. Solving this equation, we cross multiply to get x^2 = 1. The positive solution to this equation is x = 1, which confirms that the only positive number that is the same as its reciprocal is 1.
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The table shows the ages and weights of babies at a hospital.
What is the correlation between age and weight as shown in the table?
A.) none
B.) positive
C.) negative
D.) constant
Answers
Answer:
Option (b) is correct.
The correlation between age and weight shown in the table is positive.
Step-by-step explanation:
Given: The table shows the ages and weights of babies at a hospital.
We have to determine the correlation between age and weight as shown in the table.
We first Plot these points to obtain the graph as show in attachment below.
Since, Using correlation calculator , we find the correlation as 0.83113
Which is positive
Thus, The correlation between age and weight shown in the table is positive.
As age increases the weight increase.Hence, positive.
The correlation between age and weight as shown in the table is positive, This is further explained below, Option B is correct
What is correlation?Correlation is simply a relationship that exists between events or objects, A relationship between mathematical variables that are subject to variation.
In conclusion, Therefore we look at the table to decipher the correlation between age and weight.
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Write the first 4 terms of the sequence if a(4)=36 and a(n)=1/2a(n-1)+12.
A.
36, 30, 27, 25.5
B.
120, 72, 48, 36
C.
456, 216, 96, 36
D.
36, 48, 72, 120
Answers
1/2 * 3a + 12 = 36
3/2 a = 24
a = 16
so first term = 1/2 * 16 (1 - 1) + 12 = 12
which is different from any of the choices.
you sure you have the question right? Or maybe i misread it.
- Bruce added some chlorine to the water in a pool. The chlorine evaporated at a fixed rate every week... ... The table below shows the amount of chlorine f(n), in ounces, that was left in the pool after n weeks: n f(n) 1 54 2 27 3 13.5 4 6.75 Which function best shows the relationship between n and f(n)?.
Answers
To solve this problem, let us first remember that the formula for rate is:
rate = f (n) = A (r)^(n-1)
Where,
A = the 1st term
r = fractional rate of evaporation
n = number of terms
Now let us calculate for the fraction rate of evaporation, r. Since it is stated that the evaporation is happening at a fixed rate, therefore we can use any two given f (n) values to find for r. In this case, let us use when n=1 and 2.
r = 27 / 54
r = 0.5
The first term in the series is when n = 1 and that is:
A = 54
Therefore substituting all known values to the equation:
f (n) = 54 (0.5)^(n-1) (ANSWER)
If two lines are perpendicular, then they form?
Answers
hope this helps:)
What is the domain of f(x) = 3^x – 2?
Answers
Answer: In fact tis D. Is all real numbers
Step-by-step explanation: your welcome?
A binomial is defined as what?
Answers
Choose the graph below that correctly represents the equation 15x − 5y = 30.
will give brainiest to best answer!!
PLEASE HELP!!!
Answers
First, convert the equation into the form y=mx+b:
15x-5y=30
-5y=-15x+30
Divide both sides by -5.
y=-3x-6
Below is a graph of that equation:
Final answer:
To choose the correct graph that represents the equation 15x - 5y = 30, isolate y, find the slope and y-intercept, and compare to the given graphs.
Explanation:
In order to choose the correct graph that represents the equation 15x - 5y = 30, we need to solve the equation for y.
First, let's isolate y by subtracting 15x from both sides of the equation:
15x - 5y - 15x = 30 - 15x
-5y = -15x + 30
Next, divide both sides of the equation by -5 to find the value of y:
-5y/-5 = (-15x + 30)/-5
y = 3x - 6
Now that we have the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept, we can determine the graph that represents the equation. The slope of the equation is 3, which means that for every 1 unit increase in x, y increases by 3 units. The y-intercept is -6, which means that the graph intersects the y-axis at the point (0, -6).
Based on this information, we can determine that the graph that correctly represents the equation 15x - 5y = 30 is the one where the slope is 3 and it intersects the y-axis at (0, -6).
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Words that came from the same ancestral language and originated from the same word are called
Answers
In linguistics, cognates are words that have a common etymological origin. These are words in two languages that share a similar meaning, spelling, and pronunciation. For an instance, English words have lots of related words in Spanish.
The words that came from the same ancestral language and originated from the same word are called COGNATE WORDS.
What is ancestral language?In the historical linguistics tree model, a protolanguage is a hypothetical ancestral language from which several documented languages are thought to have evolved, constituting a language family. In most cases, protolanguages are unattested or, at most, only partially attested.
The languages of Norway, Sweden, Denmark, and Iceland descend from Old Norse. Numerous Indian languages, including Bengali, Hindi, Marathi, and Urdu, have their roots in Sanskrit. Each of these ancestral languages had a single common progenitor further back in time.
The term "cognates" in linguistics refers to words with the same etymological background. These are words that have the same pronunciation, spelling, and meaning in two different languages. For instance, Spanish has many words that are similar to English words.
Therefore, Words that came from the same ancestral language and originated from the same word are called COGNATE WORDS.
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The factory produces sewing machine parts. During one week, they produced 345, 298, 452 and 388 parts per day.
What was the average number of parts produced each day?
Answers
(345 + 298 + 452 + 388) / 4 = 1483/4 = 370.75 per day <=
Answer:
The produced parts average each day were 370.75.
Step-by-step explanation:
We have to find the average of parts produced by day, to calculate it we have sum all the parts produced by day and divide it by the number of days (4 days). In the equation we have:
[tex]Average=\frac{345+298+452+388}{4}=\frac{1483}{4}=370.75[/tex]
The produced parts average each day were 370.75.
the volume of a gas varies inversely as the pressure upon it. if the volume of the gas is 160 cubic centimeters at a pressure of 108 centimeters find the volume when the pressure is 87 centimeters
Answers
V α 1/P
To change it to an equality, we introduce a proportionality constant, k. We do as follows:
V = k/P
So, to determine what is asked, we need to first calculate for the value of k.
V = k/P
At V = 160 m^3 P = 108 cmHg
160 = k / (108)
k = 17280
Thus, at P = 87 cmHg
V = 17280 / 87
V = 198.62 m^3
Find the number of permutations of the first 10 letters of the alphabet taking 2 letters at a time.
Answers
10P2
=90
Therefore there are more than 90 ways in which the letters can take 2 letters at a time.
Shaun and Trudy are selling raffle tickets. So far, Shaun has sold x tickets. Trudy has sold 3 more tickets than half the number of tickets that Shaun sold. Which expression represents the number of raffle tickets that Shaun and Trudy have sold altogether?
Answers
T=s/2+3, using s from above we have:
T=x/2+3
t=T+S
t=x+x/2+3 make all term have a common denominator of 2
t=(2x+x+6)/2
t=(3x+6)/2 or if you prefer
t=1.5x+3
Answer:
3/2x+3
Step-by-step explanation:
Answer? Multiple choice?
Answers
[tex]f(x) = \frac{x^2 + 4x + 3}{x^2 - 9} [/tex]
As we know fraction is not defined where denominator is equal to zero.
So let's find where denominator is zero.
[tex]x^2 - 9 = 0[/tex]
We can factorize it as [tex]a^2 - b^2 = (a+b)(a - b)[/tex]
So we can write given equation as
[tex](x+3)(x-3) = 0[/tex]
So x = 3 or x = -3.
So function is undefined at x = 3 and x = -3.
So option 3 is correct.
Find the measures of two angles, one positive and one negative, that are coterminal with pi divided by seven.
Answers
you simply come "clockwise" to get the negative angle.
and to get the positive angle, simply go around 2π and then add π/7 to it.
Which one of the following characteristics of a solid is of concern to geometry? A. Weight B. Shape C. Color D. Substance
Answers
A house is advertised as having 1 420 square feet under roof. what is the area of this house in square meters? (a) 115 m 2
Answers
Final answer:
To convert square feet to square meters, we multiply the number of square feet (1,420) by the conversion factor (0.092903), resulting in approximately 132 square meters as the area of the house.
Explanation:
To convert the area of a house from square feet to square meters, we use the conversion factor where 1 square foot equals 0.092903 square meters. The house in question has 1,420 square feet, so we multiply this number by the conversion factor to find the area in square meters.a
1,420 square feet * 0.092903 square meters/square foot = 131.92186 square meters.
We can round the final result to the nearest whole number, which gives us approximately 132 square meters as the area of the house.
a plane intersects a prism parallel to the base of the prism. the cross section is a polygon with eight sides. The base of the prism had how many sides
Answers
Find the equation of the circle that has a diameter with endpoints located at (7, 3) and (7, –5).
Answers
The equation of the circle with a diameter between (7, 3) and (7, -5) is (x - 7)² + (y + 1)² = 2².
Explanation:To find the equation of the circle, we first need to find its center. The center of a circle is the midpoint of its diameter. To find the midpoint of (7, 3) and (7, -5), we can add the x-coordinates and divide by 2 to get 7. For the y-coordinates, we add -5 and 3 and divide by 2 to get -1. So, the center of the circle is (7, -1).
The radius of the circle is half the length of the diameter. The distance between the center of the circle and one of the endpoints of the diameter (7, 3) is the radius. Using the distance formula, the radius is √((7-7)² + (3+1)²) = 2.
Now, we can use the center and radius to write the equation of the circle, which is (x - 7)² + (y + 1)² = 2².
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A point in the form (x, y) has two dimensions. is this correct
Answers
A point has no dimension or you could say zero dimensions.
A point in the form (x, y) is simply a location on a plane.
Suppose someone opens the valve on a large water tank so that water drains out. Choose a starting volume of water in the tank (from 100 to 500 gallons), and choose how much water drains out each day (from 2 to 5 gallons). Write the equation that models the relationship between time (x) and water volume (y) in slope-intercept form.
Answers
The equation that would best represent the relationship between the given parameters is,
y = 100 - 2x
where y is the volume of water after x days.
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Help, please!
A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is 16°23'. When the boat stops, the angle of depression is 49°29' . The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round your answer to the hundredths place.
Answers
now, bear in mind that, there are 60minutes in 1 degree, so 23' is just 23/60 degrees or about 0.38 degrees.
and 29' is just 29/60 degrees or about 0.48 degrees.
so 16°23' is about 16.38°, and 49°29' is about 49.48°
[tex]\bf tan(16.38^o)=\cfrac{200}{a+b}\implies a+b=\cfrac{200}{tan(16.38^o)} \\\\\\ \boxed{b=\cfrac{200}{tan(16.38^o)}-a} \\\\\\ tan(49.48^o)=\cfrac{200}{b}\implies \boxed{b=\cfrac{200}{tan(49.48^o)}}\\\\ -------------------------------\\\\ \cfrac{200}{tan(49.48^o)}=\cfrac{200}{tan(16.38^o)}-a\\\\\\a=\cfrac{200}{tan(16.38^o)}-\cfrac{200}{tan(49.48^o)}[/tex]
make sure your calculator is in Degree mode.
The distance covered by the boat is [tex]\boxed{489.67{\text{ feet}}}.[/tex]
Further explanation:
The Pythagorean formula can be expressed as,
[tex]\boxed{{H^2} = {P^2} + {B^2}}.[/tex]
Here, H represents the hypotenuse, P represents the perpendicular and B represents the base.
The formula for tan of angle a can be expressed as
[tex]\boxed{\tan a = \frac{P}{B}}[/tex]
Explanation:
The perpendicular AB. The length of AB is [tex]200{\text{ feet}}.[/tex]
The angle of depression is [tex]\angle ACB = {14^ \circ }52'.[/tex]
One degree has 60 minutes.
[tex]{1^ \circ } = 60'[/tex]
[tex]\begin{aligned}\angle ACB &= 49 + \frac{{29}}{{60}}\\&= 49 + 0.48\\&= 49.48\\\end{aligned}[/tex]
The angle ADB is [tex]\angle ADB = {16^ \circ }23'.[/tex]
[tex]\begin{aligned}\angle ADB&= {16^ \circ }23' \\&= 16 + \frac{{23}}{{60}}\\&= {16.38^ \circ }\\\end{aligned}[/tex]
In triangle ABC.
[tex]\begin{aligned}\tan\left( {{{49.48}^\circ }} \right)&=\frac{{200}}{{BC}}\\1.13&= \frac{{200}}{{BC}}\\BC &= \frac{{200}}{{1.13}}\\BC &= 177{\text{ feet}}\\\end{aligned}[/tex]
In triangle ABD.
[tex]\begin{aligned}\tan \left( {{{16.38}^ \circ }} \right)&= \frac{{200}}{{BD}}\\0.30&= \frac{{200}}{{BD}}\\BD&= \frac{{200}}{{0.30}}\\BD &= 666.67{\text{ feet}}\\\end{aligned}[/tex]
The distance boat can travel can be obtained as follows,
[tex]\begin{aligned}DC& = BD - BC\\&= 666.67 - 177\\&= 489.67{\text{ feet}}\\\end{aligned}[/tex]
The distance covered by the boat is [tex]\boxed{489.67{\text{ feet}}}.[/tex]
Kindly refer to the image attached.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Trigonometry
Keywords: perpendicular, person watching boat, top, lighthouse, angle of depression, angle of elevation, 200 feet tall, travel, sides, right angle triangle, triangle, altitudes, hypotenuse, on the triangle, hypotenuse, trigonometric functions.
Line segment AB has a length of 3 units. It is translated 2 units to the right on a coordinate plane to obtain line segment A′B′. What is the length of A′B′?
1 unit
2 units
3 units
5 units
Answers
Answer:
It is still 3.
Step-by-step explanation:
We only translated it, so we just moved it over to the side. Translating cannot take away length.
What is the amplitude, period, and phase shift of f(x) = -3 cos (4x + pi) +6?
Answers
(x) = -3 cos(4x + π) + 6
The amplitude is |-3| = 3
If T is the period, then
(2π)/T = 4
4T = 2π
T = π/2
The period is π/2.
The phase shift is -π/4.
The vertical shift is 6.
A graph of the function is shown below.
Answer:
Amplitude = 3.
Period = π/2.
Phase shift = -π/4
The amplitude, period, and phase shift are 9, [tex]\dfrac{\pi }{2} [/tex] and [tex]\dfrac{\pi }{4} [/tex] respectively.
Periodic functionA periodic function is a function that repeats itself after a definite time interval.
Given[tex]f(x) = -3cos(4x+\pi ) +6[/tex]
The cosine function is a periodic function that repeats itself in the interval of [tex]2\pi [/tex].
Step by step explanationFor maximun amplitude
[tex]cos(4x+\pi )=-1\\ [/tex]
Then
[tex]f(x)=3+6\\ f(x)= 9[/tex]
For time period
The cosine function repeats itself in the interval of [tex]2\pi [/tex].
[tex]\begin{aligned} \frac{2\pi }{T} &= 4 \\\\ T &=\dfrac{\pi }{2}\\ \end{aligned}[/tex]
For phase shift
Phase is [tex]\dfrac{\pi }{4} [/tex]
Thus, the amplitude, period, and phase shift are 9, [tex]\dfrac{\pi }{2} [/tex] and [tex]\dfrac{\pi }{4} [/tex] respectively.
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