The safety inspector notes that Ray also needs to plan for a vertical ladder through the center of the coaster's parabolic shape for access to the coaster to perform safety repairs. Find the vertex and the equation for the axis of symmetry of the parabola, showing your work, so Ray can include it in his coaster plan.
f(x)= 3x^2+9x+12
Answers
f(x) = 3(x^2 + 3x) + 12 = 3(x+1.5)^2 +5.25
So you can see the vertex is at (0,5.25) when x = -1.5
The axis of symmetry then lies in x = -1.5
Answer with explanation:
Equation of the Parabola is
[tex]f(x)= y=3x^2+9x+12\\\\y=3[x^2+3 x+4]\\\\y=3[(x+\frac{3}{2})^2+4-(\frac{3}{2})^2]\\\\y=3[(x+\frac{3}{2})^2+(\frac{\sqrt 7}{2})^2]\\\\y-\frac{21}{4}=3[(x+\frac{3}{2})^2][/tex]
Vertex of the parabola can be obtained by
[tex]x+\frac{3}{2}=0\\\\ x=\frac{-3}{2}\\\\ y-\frac{21}{4}=0\\\\y=\frac{21}{4}\\\\ Vertex(\frac{-3}{2},\frac{21}{4})[/tex]
Axis is that line of parabola which divides the parabola into two equal halves.
[tex]x+\frac{3}{2}=0\\\\x=-\frac{3}{2}[/tex]
Related Questions
Which statement is true about the discontinuation of the function F(x)? F(x)=x+1/6x^2-7x-3
Answers
there are holes at x= 3/2 and x=-1/3
What are the vertex, focus, and directrix of the parabola with the equation x^2+8x+4y+4=0
Answers
x^2+8x+4y+4=0
4y = -x^2 - 8x - 4
y = (-1/4)x^2 - 2x - 1
To determine the vertex and the focus of the parabola, we write it in the form (y+k)^2 = x+h by completing the square method.
y + 1 = (-1/4)x^2 - 2x
y +1 = (-1/4)(x^2 + x/2)
y +1 - 1/64 = (-1/4)(x^2 + x/2 + 1/16)
y + 15/16 = (-1/4) (x + 1/4)^2
The vertex would be at point ( -1/4, -15/16)
The focus would be determined as follows:
4p=-1/4 so p=-1/8
focus = (-1/4+(-1/8),-15/16) = (-3/8,-15/16)
Directrix = x = h - p
x = -1/4 - -1/8 = -1/8
Answer:
Vertex=(-4,3)
Focus=(-4, 2)
Directrix: y=4
Step-by-step explanation:
Given the equation
[tex]x^2+8x+4y+4=0[/tex]
we have to find the vertex, focus, and directrix of parabola.
[tex]x^2+8x+4y+4=0[/tex]
[tex]y=\frac{-x^2}{4}-2x-1[/tex]
[tex]h=\frac{-b}{2a}=\frac{2}{2(\frac{-1}{4})}=-4[/tex]
Now, k can be calculated by putting x=4 and y=k in given equation
[tex]k=-1-\frac{16}{4}-2(-4)=-1-4+8=3[/tex]
The vertex is (h,k) i.e (-4,3)
[tex]y=\frac{-x^2}{4}-2x-1[/tex]
[tex]y=\frac{1}{4}(-x^2-8x-4)[/tex]
[tex]=\frac{1}{4}(-x^2-8x-16+16-4)[/tex]
[tex]y=\frac{1}{4}(-(x+4)^4+12)[/tex]
[tex]y=\frac{-1}{4}(x+4)^2+3[/tex]
which is required vertex form [tex]4p(y-k)=(x-h)^2[/tex]
gives p=-1
Focus=(-4, 3+(-1))=(-4,2)
Now directrix can be calculated as
y=3-p=3-(-1)=4
The original value of the car is $18,000 it depreciates by 15% every year what is the value of the car after three years
Answers
A=p (1-r)^t
A the value of the car after 3 years ?
P current value 18000
R rate of depreciation 0.15
T time 3 years
A=18,000×(1−0.15)^(3)
A=11,054.25
Use the graph below to answer the following question:
What is the average rate of change from x = –4 to x = 1?
–3
–1
0
1
Answers
average rate of change = (-1 - 4) / 1 - (-4) = -5 / 5 = -1
The average rate of change from x = –4 to x = 1 is 3.
Explanation:To find the average rate of change from x = –4 to x = 1, we need to calculate the change in y-values and divide it by the change in x-values. Given that the slope of the line is 3, we can use the formula for average rate of change: (change in y) / (change in x). In this case, the change in x is 1 - (-4) = 5, and the change in y is 3 * 5 = 15. Therefore, the average rate of change is 15 / 5 = 3.
Kyra is using rectangular tiles of two types for a floor design. A tile of each type is shown below: Two rectangular tiles, rectangle PQRS with vertices at P 5, 7. Q is at 9, 7. R is at 9, 12. S is at 5, 12 . Rectangle JKLM with vertices J 4, 5. K is at 6, 5. L is at 6, 10. M is at 4, 10. Which statement is correct?
Answers
The tile PQRS is bigger than tile JKLM. A rectangle is a two-dimensional shape that has two sets of equal parallel planes. Thus, its area is equal to the length multiplied by its width.
tile PQRS = (9-5)*(12-7) = 20 units²
tile JKLM = (6-4)*(10-5) = 10 units²
Tile PQRS is larger by 10 units².
Rita is saving money to buy a game. So far she has saved $15, which is three-fifths of the total of the game. How much does the game cost?
Answers
15 /3 = 5. so each 1/5 is $5
5*5 = 25
so the game costs $25
Last year the profit for a company was $560,000. This year's profit decreased by 7.1%. Find this year's profit.
Answers
Profit this year is
560,000−560,000×0.071
=520,240
C =5/9(F-32). Covert 12 degrees Celsius to Fahrenheit. Round to nearest degree
Answers
F - 32 = 12 / 5/9 = 12 * 9/5 = 21.6
F = 32 + 21.6 = 53.6 = 54 to nearest degree
The simple interest formula is I = Prt, where I represents simple interest on an amount, P, for t years at a rate of r. The equation solved for P is P = . What is the amount of money, P, that will generate $40 in interest at a 10% interest rate over 5 years? $60 $80 $90 $100
Answers
we know that
The simple interest formula is equal to
[tex]I=Prt[/tex]
where
I represents simple interest
P is the Principal amount of money to be invested
r is the rate of interest
t is Number of Time Periods
in this problem we have
[tex]t=5\ years\\ P=?\\ I=\$40\\r=0.10[/tex]
[tex]I=Prt[/tex]
Solve for P
[tex]P=I/(rt)[/tex]
substitute the values
[tex]P=40/(0.10*5)=\$80[/tex]
therefore
the answer is
[tex]\$80[/tex]
A game involves tossing a biased coin that has a 60% probability of landing on heads. If a player wins $50 when heads appears, what is the expected value of a player's winnings?
Answers
Expected value = 50(0.6) - 1(0.4) = $29.6
The interpreation of this is that if you play many times you would expect a profit of $29.6.
Find the length of the hypotenuse of a right triangle with legs of lengths 5 and 12. If necessary, round your answer to two decimal places
Answers
By the Pythagorean theorem:
c² = 5² + 12² = 25 + 144 = 169
с = √169 = 13 units ← answer
Evaluate the function rule for the given value. Y=6*4 for x=-3
Answers
y=24
Find the critical value zα/2 that corresponds to a 98% confidence level.
Answers
A critical value is the point on the scale of the test statistic (z test in this case) outside which we reject the null hypothesis, and is taken from the level of significance of the test. The critical values can be obtained from the standard distribution tables for z and for this case, it is equivalent to:
critical value zα/2 at 98% confidence level = 2.326
Answer: 2.326
Answer:
2.33
Step-by-step explanation:
Which graph correctly represents
x + 2y ≤ 4?
Answers
2y < = -x + 4
y < = -1/2x + 2
u will have a solid line (because there is an = sign in the problem)....the slope will be -1/2...so ur line is going down....u have a y int (where ur line crosses the y axis at (0,2).....u have an x int (where ur line crosses the x axis at (0,4)..ur line will be shaded below the line.
Step-by-step explanation:
[tex]x + 2y \leq 4[/tex]
To graph this inequality we replace <= symbol with = sign
[tex]x + 2y =4[/tex]
subtract x on both sides
[tex]2y =-x+4[/tex]
divide both sides by 2
[tex]y= \frac{-1}{2} x +2[/tex]
Graph the equation using a table
LEts assume some number for x and find out y
x [tex]y= \frac{-1}{2} x +2[/tex]
-2 3
0 2
2 1
Now graph the equation using points (-2,3) (0,2)(2,1)
use solid line for graphing
Now use test point (0,0) for shading
[tex]x + 2y \leq 4[/tex]
[tex]0 + 2(0) \leq 4[/tex]
[tex]0 \leq 4[/tex] true
So we shade the region that contains (0,0)
the graph is attached below
Vera and her roommates ate 1 1/3 pints of ice cream on Friday night and 1 1/6 pints of ice cream on Saturday night. How many pints did they eat in all
Answers
1 1/3 + 1 1/6
add the 1 +1 = 2
add 1/3 + 1/6 ( find common denominator, which in this case is 6)
so 1/3 becomes 2/6
2/6 + 1/6 = 3/6 which reduces to 1/2
they ate 2 1/2 pints total
I need help please :( !!!
Answers
Number of cans donated the second week = 146
Number of cans donated the third week = c, where c is some positive number
Add up the three values (132, 146, and c) to get 132+146+c. I'm choosing not to simplify because each answer choice hasn't simplified either.
The expression of 132+146+c represents the total amount of cans donated for weeks 1 through 3. We want "at least 325 cans", so that means the expression is 325 or larger. That translates to this inequality
[tex]132+146+c \ge 325[/tex]
The "greater than or equal to" sign indicates we want that sum to be 325 or larger.
Now we solve for c
[tex]132+146+c \ge 325[/tex]
[tex]278+c \ge 325[/tex]
[tex]278+c-278 \ge 325-278[/tex]
[tex]c \ge 47[/tex]
So the amount of cans donated for week three needs to be 47 or larger. If you collect exactly 47 cans for week three, then you meet the goal of 325 total cans. If you collect more than 47 cans for week three, then you exceed the goal of 325 total cans.
------------------------------------
In summary we have the inequality
[tex]132+146+c \ge 325[/tex]
which solves to
[tex]c \ge 47[/tex]
Meaning that the answer is choice D
For which value of m does the graph of y = 18x2 + mx + 2 have exactly one x-intercept?
Answers
m^2 - 4*18*2 = 0
m^2 = 144
m = +/-12
so its m = 12 or -12
Answer:
12
Step-by-step explanation:
A triangle has a perimeter of 48" and the dimensions of each side are given as X + 3, 4x-1, 2X -3 solve for the value of X and determine the length of each side
Answers
SO we can add all three sides to find perimeter of triangle.
SO perimeter is = x+3 + 4x - 1 + 2x - 3
= 7x -1
As perimeter is 48.
So we can write 7x - 1 = 48
7x - 1 = 48
On adding 1 on both sides.
7x- 1+ 1 = 48 +1
7x = 49
On dividing 7 on both sides.
[tex] \frac{7x}{7} = \frac{49}{7} [/tex]
[tex]x= 7[/tex]
So length of first side is = x+3 = 7+3 = 10
Length of second side = 4x - 1 = 4*7 - 1 = 27
Length of third side = 2x - 3 = 2*7 - 3 = 11
So side of triangle = 10 . 27 , 11
What is 2/3x = 4/15?
Answers
2/3x = 4/15 =
x = 4/15 / 2/3 = 4/15 x 3/2 = 12/30 = 2/5
x = 2/5
all i know is x=25 its hard for me to explain.
what is the radius of a circle given by the equation x^2+(y-3)^2=21
Answers
how can exponential and logarithmic functions be created to use in real world situations?
Answers
Use of Logarithmic function
We can find use of logarithmic function in measuring earthquake (Richter Scale) , the brightness of stars , and chemistry(pH balance , measure of acidity and alkalinity)
For example:
In Richter Scale , a logarithmic function that is used to measure the magnitude of earthquake.
If A = measure of amplitude of earthquake wave.
[tex]A_o[/tex] = Amplitude of smallest detectable wave.
From this we can find R the Richter scale measure of magnitude of earthquake.
[tex]R = log \frac{A}{A_o} [/tex]
Use of Exponential Function:
Exponential function may use in finding compound interest , population growth , radioactivity decay , etc.
For example :
As we know the formula for compound interest is
[tex]A = P(1+ \frac{r}{n})^{nt} [/tex]
Where P is principal amount , r = rate of interest(decimal) , n is the number of compounding period , t is time.
For
1.Find the area of each triangle with the given heights and bases
a. h = 6 inches; b = 10 inches
b. h = 9 centimeters; b = 4 centimeters
c. h = 13 yards; b = 20 yards
Answers
Answer: 1.)
A. 30in^2
B. 18cm^2
C. 130yd^2
2.) The formula for the area of a triangle is half the base times the height. So 1/2 x 40 x 32 = 640cm^2
3.) 17 yards. The fence that enclose Sondra's backyard is a right triangle whose sides measuring 8 yards, 15 yards and 17 yards respectively.
4.) From the Pythagorean Theorem we know:
hypotenuse^2 = side^2 + side^2
hypotenuse^2 = 36 + 36
hypotenuse = square root (72)
hypotenuse = 8.48528... feet
5.) a. 5
b. √128
c. √221
Area of triangle are [tex]30inches^{2}[/tex] in part(a),
[tex]18[/tex] [tex]centimerters^{2}[/tex] in part(b) and [tex]130[/tex] [tex]yards^{2}[/tex] in part(c)
What is Triangle?
In Geometry, a triangle is a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees. This property is called angle sum property of triangle.
What is area?
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.
According to questions, we have the following
In part (a.), h = [tex]6[/tex] inches; b = [tex]10[/tex] inches
In part (b.), h =[tex]9[/tex] centimeters; b =[tex]4[/tex] centimeters
In part (c.) h = [tex]13[/tex] yards; b = [tex]20[/tex] yards
We have to find the area of each triangle with the given heights and bases.
Now, Area of triangle in part a
[tex]=\frac{1}{2}[/tex]×[tex]b[/tex]×[tex]h[/tex]
[tex]=\frac{1}{2}[/tex]×[tex]6[/tex]×[tex]10[/tex]
[tex]=30inches^{2}[/tex]
Area of triangle in part b
[tex]=\frac{1}{2}[/tex]×[tex]9[/tex]×[tex]4[/tex]
[tex]=18[/tex] [tex]centimerters^{2}[/tex]
Area of triangle in part c
[tex]=\frac{1}{2}[/tex]×[tex]13[/tex]×[tex]20[/tex]
[tex]=130[/tex] [tex]yards^{2}[/tex]
Hence, we can conclude that area of triangle are [tex]30inches^{2}[/tex] in part(a),
[tex]18[/tex] [tex]centimerters^{2}[/tex] in part(b) and [tex]130[/tex] [tex]yards^{2}[/tex] in part(c)
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-13x=90-2y
-6x=48-2y
Answers
The solution to the system of equations is x = 6 and y = 84.
Explanation:The given equations are:
-13x = 90 - 2y
-6x = 48 - 2y
To find the values of x and y, we can solve the system of equations. We can start by multiplying the second equation by -2 to eliminate the y variable. This gives us:
12x = -96 + 4y
Now we have two equations:
-13x = 90 - 2y
12x = -96 + 4y
We can add the two equations together:
-13x + 12x = 90 - 2y + (-96 + 4y)
-x = -6
Then, we can divide both sides of the equation by -1 to solve for x:
x = 6
Substituting the value of x back into one of the original equations, we can solve for y. Let's use the first equation:
-13(6) = 90 - 2y
-78 = 90 - 2y
-2y = -168
y = 84
Therefore, the solution to the system of equations is x = 6 and y = 84.
This net folds into the cube shown beside it. On the cube, which letter will be on the side opposite D?
Answers
Without more information or a diagram, it's hard to give a specific answer. However, on a cube net, the face opposite 'D' is likely the one not directly connected to 'D' on the plane of the net.
Explanation:Unfortunately, without a diagram or more context, it's difficult to provide a definitive answer to the question. However, typically on a cube net, sides that are opposite each other when the net is folded into a cube are adjacent (next to each other) on the net. For example, if 'D' were on a flat square in the center of the net, the squares directly connected to it (on the top, bottom, left, and right) on the plane of the net would end up being the sides adjacent to 'D' when the cube is formed. The square not connected directly to 'D' on the plane would be opposite 'D' once the cube is formed.
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a bowl contains 80 M&M candies If 15 are orange what fraction of the candies is NOT orange. Answer reduced to lowest terms
Answers
So, they are 65/80 in the bowl that aren't orange. But we aren't done yet. We need to find the lowest term for 65/80. First, we'll need to find what can go into 65 and 80 evenly without leaving a remainder. Yes, you guessed it, five can go into 65 and 80 without leaving a remainder. So, 65 divided by 5=13 and 80 divided by 5=16. So, your final answer is 13/16.
13/16 is the final answer. This is the lowest possible term for the number pf candies that aren't orange.
65/80=13/16
13/16 are the number of M&M candies that aren't orange.
Hope I helped.
Use the change of base formula to evaluate log4 20.
Answers
so
[tex]log_a(b)=\frac{log_c(b)}{log_c(a)}[/tex]
where c is the new base
so if you wanted in base 10
[tex]log_4(20)=\frac{log_{10}(20)}{log_{10}(4)}[/tex]
using calculator, that is about 2.16096
Find the factorization of the polynomial below.
2x²+7x+6
A. (2x+2)(x+4)
B. (2x+2)(x+3)
C. (2x+3)(x+1)
D. (2x+3)(2x+2)
Answers
2x^2+7x+6
x^2+7x+12
(x+3)(x+4)
(2x+3)(x+2)
The best answer is none of the above.
When foiling the answer choices,
A: 2x^2+10x+8
B: 2x^2+8x+6
C: 2x^2+5x+3
D: 4x^2+10x+6
Answer:
(2x+3)(x+2)
Step-by-step explanation:
i hope this helps
The key idea of the transformation called a reflection is _____.
the object moves the plane a certain angle measure
the object moves in a straight line movement
the object and its image have opposite orientations
the rotating of a plane about a fixed point
Answers
Answer: The object and its image have opposite orientations
Step-by-step explanation:
A reflection is a type of rigid transformation in which the preimage is flipped across a line of reflection to produce the image, such that the figure and its image have opposite orientations.
Since, it preserves distance.
Therefore, the points of the image have the same distance from the line as the points of pre-image on the opposite side of the line.
Hence, the right option is "the object and its image have opposite orientations".
Vivian measured 2/5 cup of onions, 1/3 cup of celery, and 1/2 cup of carrots. How many cups of vegetables did Vivian measure out for her vegetable soup?
Answers
Which of the following equations represents a quadratic function?
y(y + 4)(y - 6) = 0
z2 + 2 = 3z(z2 - 1)
(2x - 3)(4x + 5) = 10x
(3b)(5b – 7)(b + 8) = 0
Answers
A quadratic equation is one in which the highest exponent of a variable is 2. We can solve this problem by expanding each choices then find which has highest exponent equal to 2.
y(y + 4)(y - 6) = 0 ---> By expansion we get y^3, therefore highest exponent is 3.
z2 + 2 = 3z(z2 - 1) ---> By expansion we get z^3, therefore highest exponent is 3.
(2x - 3)(4x + 5) = 10x ---> By expansion we get x^2, therefore highest exponent is 2.
(3b)(5b – 7)(b + 8) = 0 ---> By expansion we get b^3, therefore highest exponent is 3.
The answer to this problem is therefore:
(2x - 3)(4x + 5) = 10x