Plot the x-intercept(s), y-intercept, vertex, and axis of symmetry of this function:h(x) = (x − 1)^2− 9.
Answers
The function is
[tex]h(x)=(x-1)^2-9[/tex]1) x-intercept(s)
The x-intercepts refer to the points on which the function intercepts with the x-axis, in other words, when y=h(x)=0
So, given that condition, we get
[tex]\begin{gathered} h(x)=0 \\ \Rightarrow(x-1)^2-9=0 \\ \Rightarrow x^2-2x+1^{}-9=0 \\ \Rightarrow x^2-2x-8=0 \\ \Rightarrow(x-4)(x+2)=0 \end{gathered}[/tex]Therefore, there are two x-intercepts, and those are the points
[tex](4,0),(-2,0)[/tex]2) y-intercepts
The y-intercepts happen when x=0. So,
[tex]\begin{gathered} x=0 \\ \Rightarrow h(0)=(0-1)^2-9=1-9=-8 \end{gathered}[/tex]So, there is only one y-intercept and it's on the point (0,-8)
3) Vertex
The general equation of a parabola is
[tex]y=f(x)=a^{}x^2+bx+c[/tex]There is another way to express the same function, which is called the 'vertex form':
[tex]\begin{gathered} y=f(x)=a(x-h)^2+k \\ \Rightarrow y=ax^2-2ahx+ah^2+k \end{gathered}[/tex]What is particularly useful of this vertex form is that the vertex is the point (h,k)
So, transforming h(x) into vertex form:
[tex]\begin{gathered} h(x)=(x-1)^2-9=a(x-h)^2+k \\ \Rightarrow\begin{cases}a=1 \\ h=1 \\ k=-9\end{cases} \end{gathered}[/tex]Therefore, the vertex is the point (h,k)=(1,-9)
4) Axis of symmetry
In general, the equation of the axis of symmetry is given by
[tex]x=-\frac{b}{2a};y=f(x)=ax^2+bx+c[/tex]Therefore, in our particular problem,
[tex]\begin{gathered} h(x)=x^2-2x-8=ax^2+bx+c \\ \Rightarrow\begin{cases}a=1 \\ b=-2 \\ c=-8\end{cases} \\ \end{gathered}[/tex]Thus, the equation of the line that is the axis of symmetry is
[tex]x=-\frac{b}{2a}=-\frac{(-2)}{2\cdot1}=-\frac{(-2)}{2}=1[/tex]Then, the axis of symmetry is the line x=1.
Summing up the information in the four previous steps, we get
Related Questions
A single die is rolled twiceFind the probability of rolling a 6 the first time and a 1 the second time.
Answers
Answer:
1/36
Explanation:
In a single die, the total number of outcomes = 6
• The probability of rolling a 6 the first time = 1/6
,• The probability of rolling a 1 the second time = 1/6
Thus, the probability of rolling a 6 the first time and a 1 the second time is:
[tex]\begin{gathered} =\frac{1}{6}\times\frac{1}{6} \\ =\frac{1}{36} \end{gathered}[/tex]As people are living longer and the world's population is increasing, there are more and more people ages 65 or older. In 2000, approximately 419 million people were 65 or older. By 2017, that number increased to 656 million. On the two questions below, round to the nearest tenth when relevant a. What was the absolute change in the number of people 65 or older from 2000 to 2017? b. What was the relative change in the number of people 65 or older from 2000 to 2017?
Answers
In 2000 there were 419 million people 65 or older
In 2017 there were 656 million people 65 or older
a)
To calculate the absolute change in the number of people 65 and older you have to subtract the initial number (on year 2000) from the final number (in year 2017)
[tex]656000000-419000000=240000000[/tex]The absolute change is 240millions
b)
The relative change is the percentage of change. To calculate it you have to calculate the difference between the final number and the initial number, divide the result by the initial number and multiply it by 100
[tex]\frac{240000000}{419000000}\cdot100=57.279\cong57.30[/tex]The relative change is 57.30%
A circle has a diameter of 12 inches. Find its exact and approximate circumference and area.
Answers
STEP 1:
We write out the formulas and the necessary values
[tex]\begin{gathered} \text{Area of circle =}\pi r^2 \\ \text{circumference of circle = 2}\pi r \\ \text{radius =}\frac{diameter}{2}=\frac{12}{2}=6\text{ inches} \end{gathered}[/tex]STEP 2
We substitute the values into the formula
[tex]\begin{gathered} \text{Area of the circle = 3.14 x 6 x 6} \\ Exactvalue=113.04\text{ square inches and Approx}imate\text{ value =113} \\ \text{circumference of the circle= 2 x 3.14 x6} \\ Exactvalue=37.68\text{ inches and approx}imate\text{ value = 38inches} \\ \end{gathered}[/tex]A rectangle is bounded by the x-axis and the semicircle
y = 49 − x2 What length and width should the rectangle have so that its area is a maximum?
Answers
The length and width of the rectangle are 4.04 and 32.67 respectively for which the area is a maximum.
What is mean by Rectangle?
A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.
Given that;
The rectangle is bounded by the x - axis and the semicircle y = 49 - x².
Since,
The area of rectangle with sides x and y is,
Area = x × y
A = xy
Since, The equation of the semicircle is;
y = 49 - x².
Substitute the values of y in equation (i), we get;
A = x (49 - x²)
A = 49x - x³
Now, Find the derivative and equate into zero,
A' = 49 - 3x²
A' = 0
49 - 3x² = 0
49 = 3x²
x² = 49/3
x = 7/√3
x = 7/1.73
x = 4.04
Hence, y = 49 - x²
y = 49 - (4.04)²
y = 49 - 16.3
y = 32.67
Since, The area is maximum when we can multiply x by y as;
Maximum area = 4.04 x 32.67
Maximum area = 132
Hence, The length and width of the rectangle are 4.04 and 32.67 respectively for which the area is a maximum.
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solve the problem by defining a variable and writing an equation
Answers
Randy and Wade started riding a bike at noon. Noon is 12 pm. Both of them are heading towards each other and 60km.
let
speed of wade = x
speed of Randy = 4 + x
They met each other at 1:30 pm. 12 pm to 1:30 pm is 1 hour 30 minutes(1.5 hours). Both of them will cover a total distance of 60km.
[tex]\begin{gathered} \text{speed}=\frac{dis\tan ce}{\text{time}} \\ \text{speed}\times time=dis\tan ce \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} 1.5x+1.5(4+x)=60 \\ 1.5x+6+1.5x=60 \\ 3x=60-6 \\ 3x=54 \\ x=\frac{54}{3} \\ x=18\text{ km/hr} \end{gathered}[/tex]speed of wade = 18km/hr
speed of Randy = 4 + 18 = 22km/hr
Find a polynomial f (x) of degree 3 that has the following zeros.6 (multiplicity 2), -7Leave your answer in factored form.
Answers
If a polynomial has a zero of "a" with multilicity b, the factor would be:
[tex](x-a)^b[/tex]So, accordingly the factors would be:
[tex]\begin{gathered} (x-6)^2 \\ (x-(-7))^1 \end{gathered}[/tex]They are
[tex]\begin{gathered} (x-6)^2 \\ (x+7) \end{gathered}[/tex]We can write out the polynomial, f(x), as:
[tex]f(x)=(x-6)^2(x+7)[/tex]Evaluate theexpression belowwhen x = = 3.<54 : 2.3 - 22Enter your answer inthe box below.
Answers
The given expression is
[tex]54\frac{.}{.}2\times3-x^2[/tex]where x=3
the dot in the expression means multiplication
substitute into the expression above we have
[tex]\begin{gathered} 54\frac{.}{.}2\times3-3^2 \\ \end{gathered}[/tex]Applying BODMAS
[tex]27\times3-3^2[/tex][tex]\begin{gathered} 81-3^2 \\ 81-9 \\ 72 \end{gathered}[/tex]Therefore the value of the expression is 72
Which of the following is a proportion?8/10=6/83/4=12/154/6=9/126/9=8/12
Answers
To be able to determine which among the choices is a proportion, you check if each side has the same common ratio.
A. 8/10 = 6/8
Ratio: 0.80 = 0.75 (Not Proportional)
B. 3/4 = 12/15
Ratio: 0.75 = 0.80 (Not Proportional)
C. 4/6 = 9/12
Ratio: 0.67 = 0.75 (Not Proportional)
D. 6/9 = 8/12
Ratio: 0.67 = 0.67 (Proportional)
Since the ratio of each side of letter D is equal, Letter D is the one that is in Proportion.
#8 iOrder the figures described below according to their volumes from least (on top) to greatest (on bottom).= a cylinder with 2-inch radius and 6-inch height= a cube with side length 4 inches= a rectangular prism with a length of 2 inches, a width of 3 inches, and a height of 6 inches
Answers
step 1
Find out the volume of each figure
Cylinder
The volume of a cylinder is given by
[tex]\begin{gathered} V=\pi r^2h \\ V=(3.14)(2)^2(6) \\ V=75.36\text{ in}^3 \end{gathered}[/tex]Cube
The volume of the cube is given by
[tex]\begin{gathered} V=b^3 \\ V=4^3 \\ V=64\text{ in}^3 \end{gathered}[/tex]Rectangular prism
The volume of the prism is given by the formula
[tex]\begin{gathered} V=L*W*H \\ V=(2)(3)(6) \\ V=36\text{ in}^3 \end{gathered}[/tex]therefore
The answer is
rectangular prismcubecylinderA construction crew is lengthening a road. Let y represent the total length of the road (in miles). Let x represent the number of days the crew has worked. Suppose that x and y are related by the equation y=53 + 2x. Answer the questions below. Note that a change can be increased or a decrease. For an increase, use a positive number. For a decrease, use a negative number. What was the road’s length when the crew started working?What is the change per day in the road’s length?
Answers
Given the equation:
[tex]y=53+2x[/tex]Where y represents the total length of the road (in miles), and x represents the number of days the crew has worked.
(a) What was the road’s length when the crew started working?
When the crew started working we have x = 0. Then:
[tex]\begin{gathered} y=53+2\cdot0 \\ \therefore y=53\text{ miles} \end{gathered}[/tex]The road's length is 53 miles.
(b) What is the change per day in the road’s length?
The change per day in the road's length is 2 miles/day.
What the answer to this to solve the problem
Answers
Answer:
25
Step-by-step explanation:
180-88=92
92+61=123
123+30+x=180
153+x=180
x=25
Use the cross Products Property to solve the proportions.1. 3/4 = v/142. 5/n = 16/32
Answers
1) 3/4 = v/14
v = (3 x 14) / 4
v = 42/4
v = 10.5
2) 5/n = 16/32
5(32) = n(16)
n = 5(32) / 16
n = 160/16
n = 10
Kareem wants to attend a college that will cost $13.800 for the first year. His uncle gave him a special gift of 3000 to use toward the cost. Kareem plans to attend the college in 3 years. How much must Kareem save each month to have enough for the year cost?
Answers
Kareem needs to save $13,800 in total
He already got $3000 from his uncle.
Thus, he needs $13,800 - $3000 = $10,800 more
He will need to save up $10,800 in 3 years, that is 3 x 12 = 36 months
To find the amount he must save each month, we will divide $10,800 by 36:
[tex]\frac{10,800}{36}=\$300[/tex]Answer$300you own a pet store that sells fish tank you brought a fish tank for $35 and are going to mark it up 20% what is the selling price going to be on the fish tank
Answers
If you're marking the fish tank up by 20%, it means you're looking to sell it at 120% of its original value.
Now, let's use a rule of three to calculate such percentage:
Thereby,
[tex]x=\frac{120\cdot35}{100}\rightarrow x=42[/tex]The selling price would be $42
find the LCD of the list of fractions 7/20, 5/15
Answers
Explanation:
First we have to find multiples of each of the denominators:
[tex]\begin{cases}20\rightarrow20,40,60,80,100\ldots \\ 15\rightarrow15,30,45,60,75,90\ldots\end{cases}[/tex]From those multiples we have to find which one is the least that is in both lists. In this case, the least number that's in both lists is 60
Answer:
LCD = 60
9. ¿Cuál es el equivalente
fraccionario
de 0.2666 ...?
Answers
Answer:
4/15
Step-by-step explanation:
4/15 = 0.26666666...
what is the minimum surface area that such a box can have
Answers
Given a rectangular box with an open top and square base, the dimensions of the box are:
[tex]a\times a\times b[/tex]The volume can be calculated as:
[tex]V=a\cdot a\cdot b=a^2\cdot b[/tex]The area of the sides is:
[tex]A_L=a\cdot b[/tex]The area of the base:
[tex]A_B=a^2[/tex]There are 4 lateral sides and 1 base (the top is open), so the total surface area is:
[tex]A_{\text{total}}=4\cdot A_L+A_B=4\cdot a\cdot b+a^2[/tex]We have a fixed volume of 2048 in³, then:
[tex]\begin{gathered} a^2\cdot b=2048 \\ b=\frac{2048}{a^2} \end{gathered}[/tex]Using this result on A_total:
[tex]A_{\text{total}}=4\cdot a\cdot\frac{2048}{a^2}+a^2=\frac{8192}{a}+a^2[/tex]To find the minimum surface area, we take the derivative:
[tex]\begin{gathered} \frac{dA_{total}}{da}=-\frac{8192}{a^2}+2a=0 \\ a^3=4096 \\ a=16 \end{gathered}[/tex]Now, we calculate the minimum total area using a:
[tex]A_{\text{total}}=\frac{8192}{16}+16^2=768in^2[/tex]5|x +1| + 7 = 38
Solve for x
Answers
Answer: No solutions
Step-by-step explanation:
[tex]5|x+1|+7=-38\\\\5|x+1|=-45\\\\|x+1|=-9[/tex]
However, as absolute value is non-negative, there are no solutions.
Which of the following correctly represents the movement on the number line for the calculation 21 - (- 15) + (- 30) ?
a- left right left
b-right left left
c- right left right
d-right right left
Answers
It is the movement on the number line is right right left
The option (d) is correct .
Given,
The movement on the number line for the calculation
21 - (- 15) + (- 30)
To find the which of the following correctly represents the movement of calculation?
Now, According to the question:
21 - (- 15) + (- 30)
21 + 15 - 30 = 6
right right left
Hence, It is the movement on the number line is right right left
The option (d) is correct.
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A movie aspect ratio of 2.15:1 is shown as a letterboxed image on a TV with a width of 62.72in and a height of 35.28in what is the % of image shown on the TV
Answers
You have that the movie aspect ratio is 2.15 : 1, that is, you have following relation between width and height:
2.15/1 = 2.15 = 215%
that is, the widht is 2.15 times the height, or the width is 215% longer than height.
In order to determine what is the percentage of the image shown, you calculate the percentage that widht is more longer than height. You have a TV of 62.72 width and 35.28 in height:
62.72/35.28 = 1.77 = 177%
that is, width of TV is 1.77 times longer than height, or width is 177% longer.
Hence, on TV will be not possible to watch the complete image. And the percentage shown is of 177%.
A recipe uses 6 cups of flour to 1 1/10 cups of milk. If you have 2 cups of flour, how much milk should you use?
Answers
We were told that the recipe uses 6 cups of flour to 1 1/10 cups of milk. Converting 1 1/10 to improper fraction, it becomes 11/10
Let x represent the number of cups of milk that would be used for 2 cups of flour. The equations would be as shown below
11/10 = 6
x = 2
By cross multiplying, we have
2 * 11/10 = 6 * x
6x = 22/10 = 11/5
x = (11/5) / 6
If we flip 6 such that it becomes 1/6, the division sign changes to multiplication. Thus, we have
x = 11/5 * 1/6 = 11/30
Thus, 11/30 cup of milk should be used
Perform a DuPont analysis on Healthy Body Nursing Home, Inc. Assume that the industry average ratios are as follows: Total margin: 3.9%
Total asset turnover: 0.5
Equity multiplier: 2.8
Return on equity: %
Answers
Using the DuPont analysis the return on equity is 5.46%.
What is the return on equity?
Return on equity is the ratio of net income to average total equity. Return on equity is an example of a profitability ratio. Profitability ratios measure the ability of a firm to generate profits using available resources.
Return on equity = net income / average total equity
Using the Dupont formula, return on equity can be determined using:
Return on equity = total margin x asset turnover x equity multiplier
Return on equity = (Net income / Sales) x (Sales/Total Assets) x (total asset / common equity)
3.9% x 0.5 x 2.8 = 5.46%
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you bought a car for $5000. each year it depreciates by 8.5%. Which equation can be used to find the value, v, of the car, x years after it was purchased?
Answers
We have the following:
In this case, we have the following formula:
[tex]v=C\cdot(1-r)^x[/tex]Where C is the original value of the car, r is the depreciation rate and x is the time in years
Write 3.25% as a fraction in simplest form. Can you explain step by step please?
Answers
From the problem, we have 3.25% to convert into fraction.
Note that percentage a% can be written as a/100
So 3.25% will be :
[tex]3.25\%=\frac{3.25}{100}[/tex]From here, we can multiply the numerator and the denominator by 100 to make 3.25 a whole number.
[tex]\frac{3.25\times100}{100\times100}=\frac{325}{10000}[/tex]Next step is to simplify the fraction by getting the factors of the numerator and the denominator.
325 = 25 x 13
10000 = 25 x 400
Rewriting it again :
[tex]\frac{325}{10000}=\frac{25\times13}{25\times400}[/tex]Cancel the common factor (25)
[tex]\frac{\cancel{25}\times13}{\cancel{25}\times400}=\frac{13}{400}[/tex]The answer is 13/400
Help me out please I don’t understand what I’m doing
Answers
Since we have the value for selling each shirt, the earnings that came from the hats sold and the total earnings we can complete equation using a linear equation in which the cost of each shirt will represent the slope and the y-intercept will be the earnings that came from the hats, like this:
[tex]5x+40=125[/tex]then clear the equation for x in order to find how many shirts were sold
[tex]\begin{gathered} 5x=125-40 \\ 5x=85 \\ x=17 \end{gathered}[/tex]Need help with this question
Answers
Given: a quadratic function with vertex (2,3) opening upward .
Find: the given statement is true or false.
Explanation: if parabola has a vertex at (2,3) and opens upward, it has one real solution., (2,3) will be a lowest point. The vertex will be at lowest point, it will be minimum.
that means graph has no one real solution. hence it will never going to intersect. so this statement is false.
Final answer: the given statement is FALSE.
Write the equation for a parabola with a focus at (1,-4) and a directrix at x= 2.x=?
Answers
The basic form of the equation is;
4p (x- h)= (y - k)²
where (h, k) is the vertex and p is the distance from the vertex to either of its directrix or the focus
But, focus is = (1, -4) adn directrix = 2
So, the perpendicular point is :
(1.5 , -4)
p = -0.5
Putting all the values into the formula
4p (x- h)= (y - k)²
4(-0.5)(x - 1.5) = (y - (-4)²
simplify
-2(x - 1.5) = (y + 4)²
-2(x - 1.5) = y² + 8y + 16
Divide through the equation by -2
x - 1.5 = (-1/2) y² - 4y - 8
Add 1.5 to both-side of the equation
[tex]x\text{ = -}\frac{1}{2}y^2-4y\text{ - 8 + 1.5}[/tex][tex]x=-\frac{1}{2}y^2-4y\text{ - 6.5}[/tex]Hi. Been out due to medical issues trying to catch-up and learn my work. Thank you in advance
Answers
The break even point is where cost is equal to revenue
cost : y = 25.96x + 22752
Revenue : y = 57.56x
Set the two equations equal to determine x
25.96x + 22752 = 57.56x
Subtract 25.96x from each side
25.96x-25.96x + 22752 = 57.56x-25.96x
22752 = 31.6x
Divide each side by 31.6
22752/31.6 = 31.6x/31.6
720 =x
Now find the value of y
y = 57.56x
y = 57.56 (720)
y = 41443.20
( 720, 41443.20)
Answer: ( 720, 41443.20)
The radius, R, of a sphere is 5.7 cm. Calculate the sphere's volume, V.Use the value 3.14 for r, and round your answer to the nearest tenth. (Do not round any intermediate computations.)
Answers
The volume formula of a sphere is :
[tex]V=\frac{4}{3}\pi r^3[/tex]From the problem, the radius of the sphere is r = 5.7 cm
Using the formula above :
[tex]\begin{gathered} V=\frac{4}{3}(3.14)(5.7)^3 \\ V=775.3 \end{gathered}[/tex]ANSWER :
775.3 cm^3
In the circle below, if AB is a diameter, find the measure of arc AB.
Answers
A circle has 360°
since the diameter cuts in half the circle
The measure of the arc of half the circle (semicircle) measure arcAB is
[tex]AB=\frac{360}{2}[/tex][tex]AB=180[/tex]AB=180°
Then correct answer is
option a