Answer:
[tex]\textsf{Vertex form}: \quad y=\dfrac{1}{36}(x+3)^2+1[/tex]
[tex]\textsf{Standard form}: \quad y=\dfrac{1}{36}x^2+\dfrac{1}{6}x+\dfrac{5}{4}[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{5.6 cm}\underline{Vertex form of a quadratic equation}\\\\$y=a(x-h)^2+k$\\\\where:\\ \phantom{ww}$\bullet$ $(h,k)$ is the vertex. \\ \phantom{ww}$\bullet$ $a$ is some constant.\\\end{minipage}}[/tex]
Given:
Vertex = (-3, 1)Point = (-9, 2)Substitute the given vertex and point into the formula and solve for a:
[tex]\begin{aligned}y&=a(x-h)^2+k\\\\\implies 2&=a(-9-(-3))^2+1\\2&=a(-6)^2+1\\2&=36a+1\\36a&=1\\a&=\dfrac{1}{36}\end{aligned}[/tex]
Therefore, the equation of the parabola in vertex form is:
[tex]\boxed{y=\dfrac{1}{36}(x+3)^2+1}[/tex]
In standard form:
[tex]\implies y=\dfrac{1}{36}(x+3)(x+3)+1[/tex]
[tex]\implies y=\dfrac{1}{36}(x^2+6x+9)+1[/tex]
[tex]\implies y=\dfrac{1}{36}x^2+\dfrac{6}{36}x+\dfrac{9}{36}+1[/tex]
[tex]\implies y=\dfrac{1}{36}x^2+\dfrac{1}{6}x+\dfrac{5}{4}[/tex]
How do you know when an equation has one solution? A. The coefficients are differentB. The coefficients are the same and the constants are differentC. The coefficients are the same and the constants are same
SOLUTION
Given the question in the question tab, the following are the solution steps to answer the question.
STEP 1: Define an equation with one solution
You can tell that an equation has one solution if you solve the equation and get a variable equal to a number. A system of linear equations has one solution when the graphs intersect at a point.
STEP 2: Examine the options
An equation is said to have "no solution" when the coefficients are the same and the constants are different as seen below:
[tex]\begin{gathered} 4x+12=24+4x \\ 4x-4x=24-12 \\ 0=12 \end{gathered}[/tex]When the coefficients are the same and the constants are the same, the equation is said to have "infinitely many solutions as seen below:
[tex]\begin{gathered} 4x-14=-6-8+4x \\ 4x-14=-14+4x \\ 4x-4x=-14+14 \\ 0=0 \end{gathered}[/tex]When the coefficients are different, the equation is said to have one solution as seen below:
[tex]\begin{gathered} 2x+56=3x-12 \\ 2x-3x=-12-56 \\ -x=-68 \\ x=68 \end{gathered}[/tex]Hence, the answer is:
"When the coefficients are different"
= 186x – 30y5х – 25y= 15
Answer:
There are infinitely many solutions
Explanation:
Given the system of linear equations:
[tex]\mleft\{\begin{aligned}6x-30y=18 \\ 5x+25y=15\end{aligned}\mright.[/tex]To solve using Matrices, we write the equation in the form:
Ax = b
Where A represent the matrix of the coefficient of the variables x and y
x represent the matrix of the variables x and y
and b represents the matrix of the constants on the right hand side.
In matrix form, we have:
[tex]\begin{bmatrix}{6} & {-30} \\ {5} & {-25}\end{bmatrix}\begin{bmatrix}{x} \\ {y}\end{bmatrix}=\begin{bmatrix}{18} \\ {15}\end{bmatrix}[/tex]We must have a nonzero determinant for A
The determinant of A is:
[tex]\begin{gathered} (-25\times6)-(-30\times5) \\ =-150+150 \\ =0 \end{gathered}[/tex]The determinant of A is zero, without going far, we conclude that the system has infinite number of solutions.
Write the correct inequality for the following statement and then solve for the given number of
guests.
You are in charge of planning your mother's surprise party. You have
found a hall that rents for $225 and a caterer that will charge you $15 per
guest. Write an inequality to calculate your total cost (T) based on the
number of guests (g).
If you have a total budget of $1000 how many guests can you
invite?
The inequality that can be used to calculate total cost (T) based on the number of guests (g) is 225 + 15g ≤ 1000.
If you have a total budget of $1000, 51.67 guests can be invited.
How to write inequality equation?Cost of hall rent = $225Caterer's charge per guest = $15Total cost = TNumber of guests = gTotal cost = Cost of hall rent + Caterer's charge per guest(Number of guests)
225 + 15g ≤ T
If you have a total budget of $1000
225 + 15g ≤ T
225 + 15g ≤ 1000
substract 225 from both sides15g ≤ 1000 - 225
15g ≤ 775
divide both sides by 15g ≤ 775 / 15
g ≤ 51.66666666666666
Approximately,
g ≤ 51.67
So therefore, the inequality to represent the situation is 15g ≤ 1000 - 225 and about 51.67 guests can be invited if you have a budget of $1000
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Bobby can work at most 20 hours next week, but she needs to earn at least $150. Herdog-walking job pays her $6 per hour and her iob as a car wash attendant pays her $10 perhour. If d is the number of hours she walks dogs this week and w is the number of hours sheworks at the car wash, write a system of inequalities that represents Bobby's situation.
Solution:
Let d represent the number of hours Bobby walks dogs,
and w be the number of hours she works at the car wash.
Given that she can work at most 20 hours next week, this implies that the total number of hours she works is expressed as
[tex]d+w\leq20[/tex]If her dog-walking job pays her $6 per hour, this implies that for d number of hours, she will earn
[tex]\begin{gathered} \frac{\$6}{1\text{ hour}}\times d\text{ hours} \\ =\$\text{ 6d} \end{gathered}[/tex]and her iob as a car wash attendant pays her $10 per hour, similarly for w number of hours, she will earn
[tex]\begin{gathered} \frac{\$10}{1\text{ hour}}\times w\text{ hours} \\ =\$10w \end{gathered}[/tex]Given that she needs to earn at least $150, this implies that her total earnings will be expressed as
[tex]6d+10w\ge150[/tex]Hence, the system of inequalities that represents Bobby's situation is:
[tex]\begin{gathered} d+w\leq20 \\ 6d+10w\ge150 \end{gathered}[/tex]what is the exact value of tan (-3pi/4)
Answer:
1
Step-by-step explanation:
Your phone plan charges you an initial fee, and then a certain amount depending on the amount of data you use. They send you periodic updates of your usage andyour current bill cost. After using 3 GB of data you owe $30. After 5 GB of data, you owe $40. What is the initial fee prior to the data usage charge?A.$15B.$5C.$2.50D.$10
If for 3 Gb we get a bill for $30 and for 5 Gb we get a bill of $40. It implies that every Gb costs $5, therefor the equation we have use 3 Gb is
[tex]30=3\cdot5+b[/tex]where b is the initial fee. Solving this for b we get that
[tex]b=30-15=15[/tex]so the initial fee is $15
Write a rational expression to determine how many cubbies high the book shelf will be.
Answer:
[tex]\frac{x^2-8x-65}{x+1}[/tex]
Explanation:
The height of each small division is x + 1, and the total height is x² + 6x + 5. Then, the number of small division is the total height divided by the small division height.
Which expression is equivalent to -5/3 x
Answer:
-1 2/3 x
Step-by-step explanation:
This is not that difficult. But I suggest that you provide answer options.
pls mark brainliest
Find the distance between each pair of points. Round youranswer to the nearest tenth, if necessary.(7, 6), (0, 2)
Chris, this is the solution to the exercise:
• Point 1 = (7, 6)
,• Point 2, (0, 2)
Now, let's use the formula to calculate the distance between the two points, as follows:
d = √(0 - 7)² + (2 - 6)²
d = √-7² + (-4)
d = √49 + 16
d = √65
d = 8.0622
d = 8.1 (rounding to the nearest tenth)
Reflect (1, -3) over the y-axis.
Then translate the result down 1 unit.
What are the coordinates of the final point?
Answer:
(-1 , -4)
Step-by-step explanation:
Hello!
Refer to the graph for the reflection and translation.
The coordinates of the final point would be (-1,-4), given the x and y-values of the point.
11. It takes 4 hours to type 8 memos. How long will it take to type 56 memos?
Answer:28
Step-by-step explanation:if the unknown hour is x
then it would be
4hr=8memos
x hr=56 memos
cross multiply that would be
4×56÷8=28
and to check this answer you can find how many hours it would take to write one memo and multiply it by 28(your answer) that would be
x hr=1 memo
4hr =8 memo
cross multiply that would be
8÷4=
and then check it by doing 28 × 2=56
The time taken to type 56 memos will be 28 hours.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that it takes 4 hours to type 8 memos. The time taken to type 56 memos will be calculated as,
8 memos = 4 hours
56 memos = ( 56 x 4 ) /8
56 memos = 28 hors
Therefore, it will take 28 hours to type 56 memos.
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Which equation is equivalent to −(5−x)=5−x? Responses −5−x=5−x − 5 − x = 5 − x −5+x=5−x − 5 + x = 5 − x 5−x=5−x 5 − x = 5 − x 5+x=5−x
The equation is equivalent to −(5−x) = 5−x is -5+x = 5-x
The given equation −(5−x) = 5−x
The distributive property states that multiplying the sum of two or more variables by a number gives the same result as when each variable is multiplied individually by the number and the products are added together.
The distributive property of addition
A(B+C) = AB + AC
The distributive property of subtraction
A(B-C) = AB - AC
The given equation is −(5−x) = 5−x
Apply distributive property in the equation
−(5−x) = 5−x
-5+x = 5-x
Hence, The equation is equivalent to −(5−x) = 5−x is -5+x = 5-x
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8.1÷0.9+0.4^(3)÷0.8×30
Answer:
11.4
Step-by-step explanation:
Answer:
11.4
p.s you try it may help
Dean started with 5 metersof yarn. He used 300millimeters of yarn each dayfor 5 days. How manymeters of yarn does Deanhave left?
Given:
Dean started a yarn with 5 meters. He used 300millimeters of yarn each day for 5 days.
Required:
Find how many meters of yarn Dean has left.
Explanation:
We know that 1 meter = 1000milimeters
Dean used meters of yarn in the five days
[tex]\begin{gathered} =\frac{300\times5}{1000} \\ =\frac{3}{2}\text{ meter} \end{gathered}[/tex]The left meters are:
[tex]\begin{gathered} =5-\frac{3}{2} \\ =\frac{10-3}{2} \\ =\frac{7}{2} \\ =3.5 \end{gathered}[/tex]Final Answer:
Thus 3.5 meters of yarn left
PLEASE HELP
A bag contains 42 red, 45 green, 20 yellow, and 32 purple candies. You pick one candy at random, find the probability that it is yellow or green.
p(Yellow or not green)= ?
Answer:
I think the answer is 94/139
Step-by-step explanation:
Answer:
65/139
Step-by-step explanation:
Total number of candies - 139
No of yellow and green - 65
Thus probability - 65/139
i need help with this question
In the given expression
[tex]x^5+32y^3=\ln y[/tex]To differentiate it with respect to it, we will writ y' for every term of y
The differentiation of
[tex]x^5=5x^{5-1}=5x^4[/tex]The differentiation of
[tex]32y^3=32(3)y^{3-1}y^{\prime}=96y^2y^{\prime}[/tex]The differentiation of
[tex]\ln y=\frac{1}{y}(y^{\prime})=y^{-1}y^{\prime}[/tex]Now let us write all of them in one line
[tex]5x^4+96y^2y^{\prime}=y^{-1}y^{\prime}[/tex]Put the terms of y' on one side and the other term on the other side
[tex]96y^2y^{\prime}-y^{-1}y^{\prime}=-5x^4[/tex]Take y' as a common factor
[tex]y^{\prime}(96y^2-y^{-1})=-5x^4[/tex]Divide both sides by the bracket
[tex]y^{\prime}=\frac{-5x^4}{(96y^2-y^{-1})}[/tex]Since the given point is (-2, 1), then
x = -2 and y = 1
Substitute them to find y'
[tex]\begin{gathered} y^{\prime}=\frac{-5(-2)^4}{(96\lbrack1\rbrack^2-\lbrack1\rbrack^{-1})}=\frac{-5(16)}{(96-1)}=\frac{-80}{95} \\ y^{\prime}=-\frac{16}{19} \end{gathered}[/tex]The value of y' is -16/19
The ------------- of a circle is a line segment that joins two points on the circumference of the circle and passes through the centre of the circle
Solution
The Diameter of a circle is a line segment that joins two points on the circumference of the circle and passes through the centre of the circle
Find the surface area of a cylinder with a height of 7 in and a base diameter of 8 in.
Use the value 3.14 for pi and do not do any rounding.
Be sure to include the correct unit.
The total surface area of the cylinder whose height is 7 units and diameter is 8 units is 226.08 units.
What is surface area?
The surface area of a solid item is a measurement of the total volume that the surface of the thing occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more complicated than the definition of arc length for one-dimensional curves or the definition of surface area for polyhedra (i.e., objects with flat polygonal faces), where the surface area is equal to the sum of the areas of its faces. When smooth surfaces, like spheres, are represented as parametric surfaces, their surface area can be calculated. This definition of surface area, based on infinitesimal calculus methods, uses partial derivatives and double integration.
The formula to calculate the total surface area of a cylinder is:
[tex]S = 2\pi rh + \pi r^2[/tex]
Where, S = surface area, r = radius, and h = height
As given in the question,
height (h) = 7 units
diameter (d) = 8 units
Since, r = d/2
So, r = 8/2 = 4 units
putting these values in the formula:
[tex]S = (2\times 3.14 \times 4 \times 7)+( 3.14 \times 4^2)[/tex]
[tex]S = (2\times 3.14 \times 4 \times 7)+( 3.14 \times 16)[/tex]
[tex]S = 175.84 + 50.24\\[/tex]
S = 226.08 units
Therefore, Surface area of the cylinder is 226.08 units
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Factor the following expression completely.x4 - 16
Given:
[tex]x^4-16[/tex]Required:
To factor the given expression.
Explanation:
Consider the given expression
[tex]x^4-16[/tex]Now
[tex]\begin{gathered} x^4-16=(x^2)^2-(4^)^2 \\ \\ =(x^2-4)(x^2+4) \\ \\ =(x^2-2^2)(x^2+4) \\ \\ =(x-2)(x+2)(x^2+4) \end{gathered}[/tex]Final Answer:
[tex]x^4-16=(x-2)(x+2)(x^2+4)[/tex]Simplify the following exponent (x¹⁰ ÷ x²) x x³
Answer:
x^6
Step-by-step explanation:
Given X¹⁰ ÷ (x² X x³ ÷ x) : Simplify.
We know the law of exponents which states that
a^m x a^n = a^m + n
a^m / a^n = a^m - n
So the problem is x^10 / (x^2 x x^3 / x)
So we have x^10 / x ^5 / x
x ^10 / x ^5 - 1
x^10 / x^4
x ^10 -
x^6
After simplifying we get x^6
Together, Nora and Addison own 152 shares of stock. Nora has 4 less than three times Addison's amount. How many shares does each girl have?
The number of shares that Nora has is 113.
The number of shares that Addison has is 39.
How many shares do they have?Let x represent the amount of shares that Addison has
Shares that Nora has = 3x - 4
The sum of shares they both have : 3x - 4 + x = 152
In order to determine the value of x, take the following steps:
Combine similar terms together: 3x + x = 152 + 4
Add similar terms: 4x = 156
Divide both sides by 4 : x = 156/ 4 = 39
Shares that Nora has = 3(39) - 4
117 - 4 = 113
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Write an equivalent expression for (x+2)+(x+2)
We will proceed as follows in order to get an equivalent expression:
[tex](x+2)+(x+2)=(x+x)+(2+2)=2x+4[/tex]So, one equivalent expression is 2x + 4.
Find the mean of these numbers: 5, 11, 2, 12, 4, 2Need solution :D
SOLUTION
We want to find the mean of the numbers 5, 11, 2, 12, 4, 2
To do this, we add the numbers and divide by the total numbers.
The numbers are 6 in numbers, so the mean
[tex]\begin{gathered} mean=\frac{5+11+2+12+4+2}{6} \\ =\frac{36}{6} \\ =6 \end{gathered}[/tex]Hence the answer is 6
PLS HELP I HAVE LESS THAN 20 MINUTES!!!!
The statement "Samuel will get 25 or more for mowing the lawn" can be represented using the inequality → x ≥ 25 and the graph number [2] represents the correct graph.
What is an inequality?In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. There are two types of inequalities namely strict and slack inequalities.
Given is the following statement - "Samuel will get 25 or more for mowing the lawn".
The given statement is -
Samuel will get $25 or more for mowing the lawn.
Assume that Samuel gets $x for mowing the lawn.
Then according to the question, we can write the inequality as -
x ≥ 25
" ≥ " sign to represent either 25 or more.
So, the inequality can be written as x ≥ 25 and the graph number [2] represents the correct graph.
Therefore, the statement "Samuel will get 25 or more for mowing the lawn" can be represented using the inequality → x ≥ 25 and the graph number [2] represents the correct graph.
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Hiro picks up four digits card, he says: I can make a multiple of 10 with these cards. I can also make a number that rounds to 5300 and a number where the digit 8 is worth 800?
PLEASE AWNSER NOW Sandy uses three and one half pounds of clay to make 7 small vases. Use a proportion to find how many pounds of clay she will use to make 21 small vases.
7 and one fourth pounds
8 and one half pounds
10 and one half pounds
14 and one fourth pounds
Answer:
The answer is 10 and one half pounds
Explanation:
3.5/7 = 2
21/2 = 10.5
10 and one half pounds use to make 21 small vases.
What is Unitary Method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.
For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.
12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.
As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.
This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.
Given:
Sandy uses clay= 3 1/2 pounds for 7 vase
So, for 1 vase amount of clay used
= 7/2 x 1/7
= 1/2 pounds
Now, for 21 small vases amount of clay used
=21 x 1/2
= 21/2
= 10 1/2 pounds.
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NEED HELP ASAP
The graph compares the ages of 100 randomly selected visitors at a beach in January and in July. Select from the drop-down menu to correctly complete the statement. On average, visitors in January were Choose... years older than visitors in July.
From the given box and whisker plot, On average, visitors in January were 10 years older than visitors in July.
How to Interpret Box and Whisker Plot?A box and whisker plot is a graph plot that indicates whether a distribution is skewed and whether there are potential unusual observations (outliers) in the data set.
Now, the way to interpret it is as follows;
The left side of the box is the lower quartile. The right side of the box is the upper quartile. The box covers the interquartile interval, where 50% of the data is found.The vertical line that split the box in two is referred to as the median. The whiskers are the two lines outside the box, that go from the minimum to the lower quartile and then from the upper quartile to the maximum.Now, looking at the box and whisker plot for January and July, we can say that those in January are approximately 10 years older than those in July.
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Describe the graph of the quadratic function f (x) = -3x^2+ 6x + 5 by stating whether it opens upward or downward and stating how it is stretched or compressed when compared to the parent quadratic functionA. The parabola opens downward. It is compressed shorter by a factor of 3.B. The parabola opens upward. It is stretched taller by a factor of 3.C. The parabola opens downward. It is stretched taller by a factor of 3.D. The parabola opens upward. It is compressed shorter by a factor of 3.
Answer:
B. The parabola opens upward. It is stretched taller by a factor of 3.
Step-by-step explanation:
You want to know how the quadratic opens, and how it is stretched or compressed, given its equation is f(x) = -3x^2 +6x +5.
Leading coefficientThe "leading coefficient" is the coefficient of the highest-degree term. That term is -3x^2, and its coefficient is -3.
The minus sign tells you the parabola opens downward.
The magnitude of 3 tells you the parabola is stretched vertically (taller) by a factor of 3.
What's 2,412 Divided by 25
Answer:
96.48
Step-by-step explanation:
The answer is 96.48.
Answer:
96.48
Step-by-step explanation:
2,412 divided by 25 equals 96.48
Simplify completely (7x²+11x-6)/(7x²-10x+3)
One way of simplifying the given expression is by factoring them, and then simplifying the common terms.
So, we need to write:
[tex]\begin{gathered} 7x^{2}+11x-6=7(x-a)(x-b) \\ \text{and} \\ 7x^{2}-10x+3=7(x-c)(x-d) \end{gathered}[/tex]The constants a and b are the zeros of the first expression, and the constants c and d are the zeros of the second expression.
So, we can find those zeros using the quadratic formula. We obtain, for the first expression:
[tex]\begin{gathered} x=\frac{-11\pm\sqrt[]{11^{2}-4(7)(-6)}}{2(7)} \\ \\ x=\frac{-11\pm\sqrt[]{289}}{14} \\ \\ x=\frac{-11\pm17}{14} \\ \\ a=\frac{-11-17}{14}=-2 \\ \\ b=\frac{-11+17}{14}=\frac{6}{14}=\frac{3}{7} \end{gathered}[/tex]And, for the second expression, we obtain:
[tex]\begin{gathered} x=\frac{-(-10)\pm\sqrt[]{(-10)^{2}-4(7)(3)}}{2(7)} \\ \\ x=\frac{10\pm\sqrt[]{16}}{14} \\ \\ x=\frac{10\pm4}{14} \\ \\ c=\frac{10-4}{14}=\frac{6}{14}=\frac{3}{7} \\ \\ d=\frac{10+4}{14}=1 \end{gathered}[/tex]Then, we can write:
[tex]\begin{gathered} 7x^2+11x-6=7(x-(-2))(x-\frac{3}{7})=7(x+2)(x-\frac{3}{7}) \\ \\ 7x^2-10x+3=7(x-\frac{3}{7})(x-1) \end{gathered}[/tex]Thus, the given function can be simplified as follows:
[tex]\frac{7x²+11x-6}{7x²-10x+3}=\frac{7(x+2)(x-\frac{3}{7})}{7(x-\frac{3}{7})(x-1)}=\frac{x+2}{x-1}[/tex]Therefore, the answer is:
[tex]\mathbf{\frac{x+2}{x-1}}[/tex]