Answer:
The vertex form is y = -3(x + 2)² + 10
Step-by-step explanation:
* Lets revise how to put the quadratic in the vertex form
- The general form of the quadratic is y = ax² + bx + c, where
a , b , c are constants
# a is the coefficient of x²
# b is the coefficient of x
# c is the numerical term or the y-intercept
- The vertex form of the quadratic is a(x - h)² + k, where a, h , k
are constants
# a is the coefficient of x²
# h is the x-coordinate of the vertex point of the quadratic
# k is the y-coordinate of the vertex point of the quadratic
- We can find h from a and b ⇒ h = -b/a
- We find k by substitute the value of h instead of x in the general form
of the quadratic
k = ah² + bh + c
* Now lets solve the problem
∵ y = -3x² - 12x - 2
∵ y = ax² + bx + c
∴ a = -3 , b = -12
∵ h = -b/2a
∴ h = -(-12)/2(-3) = 12/-6 = -2
- Lets find k
∴ k = -3(-2)² - 12(-2) - 2 = -3(4) + 24 - 2 = -12 + 24 - 2 = 10
* Lets writ the vertex form
∵ y = a(x - h)² + k
∵ a = -3 , h = -2 , k = 10
∴ y = -3(x - -2)² + 10
∴ y = -3(x + 2)² + 10
* The vertex form is y = -3(x + 2)² + 10
ANSWER
The vertex form is:
[tex]y = - 3{(x + 2)}^{2} +10[/tex]
EXPLANATION
The given equation is:
[tex]y = - 3 {x}^{2} - 12x - 2[/tex]
[tex]y = - 3( {x}^{2} + 4x) - 2[/tex]
We add and subtract the square of half the coefficient of x.
[tex]y = - 3( {x}^{2} + 4x + {2}^{2} ) - - 3( {2})^{2} - 2[/tex]
We simplify to get,
[tex]y = - 3{(x + 2)}^{2} + 3(4)- 2[/tex]
[tex]y = - 3{(x + 2)}^{2} +10[/tex]
This is in the vertex form.
The lengths of two sides of a right triangle are given. Find the length of the third side. Round to the nearest tenth if necessary.
leg: 20 m; hypotenuse: 25 m
10.5 m
8.9 m
32 m
15 m
Using the Pythagorean theorem, given a right triangle with one leg 20 m and the hypotenuse 25 m, the length of the other leg is approximately 15 m.
Explanation:To find the length of the third side of a right triangle, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In other words, if the lengths of the two legs are a and b, and the hypotenuse is c, the theorem is expressed as a^2 + b^2 = c^2.
Since we know the length of the hypotenuse (25 m) and one leg (20 m), we need to find the length of the other leg. Let's denote it as b. Therefore, the equation will be 20^2 + b^2 = 25^2. Solve for b, we get b = sqrt(25^2 - 20^2), which equals approximately 15 m.
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match the expression please!!! 20 points
Answer:
r
0.15r
1.15r
Step-by-step explanation:
The problem states that r cm of rain fell on the garden on Thursday, so the first answer is:
The centimeters of rain on Thursday r
On Saturday, 15% more rain fell than on Thursday. The extra amount of rain on Saturday, as compared to Thursday is 15% of r. 15% of r is the same as 15% times r. 15% can be written as a decimal as 0.15, so 15% of r is the same as 0.15r. This is the answer to the second line.
How many more centimeters of rain fell on Saturday than on Thursday 0.15r
The amount of rain on Saturday is the amount of rain on Thursday plus the additional 15%. We already know that the additional 15% of rain is 0.15r. The total amount of rain on Saturday is r + 0.15r = 1r + 0.15r = 1.15r. That answers the third question.
The centimeters of rain on Saturday 1.15r
what is the equation of a line with a slope of -2 that passes the point (6,8)
Answer:
y - 8 = -2(x - 6)
Step-by-step explanation:
Given the slope and one point on the line, we should use the point-slope formula:
y - k = m(x - h), where (h, k) is the point and m is the slope.
Here, we have y - 8 = -2(x - 6)
Answer:
y = - 2x + 20
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
here m = - 2, so
y = - 2x + c ← is the partial equation
To find c substitute (6, 8) into the partial equation
8 = - 12 + c ⇒ c = 8 + 12 = 20
y = - 2x + 20 ← equation of line
Simplify the expression using long division. (10x2 – 85x – 10) ÷ (x – 8)
Answer: 5 • (2x2 - 17x - 2)
———————————————————
x - 8
Step-by-step explanation:
Step by step solution :
Step 1 :
Equation at the end of step 1 :
Step 2 :
10x2 - 85x - 10
Simplify ———————————————
x - 8
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
10x2 - 85x - 10 = 5 • (2x2 - 17x - 2)
Trying to factor by splitting the middle term
3.2 Factoring 2x2 - 17x - 2
The first term is, 2x2 its coefficient is 2 .
The middle term is, -17x its coefficient is -17 .
The last term, "the constant", is -2
Step-1 : Multiply the coefficient of the first term by the constant 2 • -2 = -4
Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is -17 .
-4 + 1 = -3
-2 + 2 = 0
-1 + 4 = 3
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
11-4x<-1 can someone help me asap
-4x<-12
-4x<-12 | /(-4)
x>3
Complete the parametric equations for the given rectangular equation.
Answer:
1) x = 5√(1-sin^2t)
2) y = = -2+3t
Step-by-step explanation:
1)
Given rectangular equation:
x^2/25 + y^2 =1
Parametric equations y=-sint and x=?
Find parametric equation for x
Substituting value of y =-sint in given rectangular equation, we get
x^2/25 + (-sint )^2 =1
x^2/25 + sin^2t =1
x^2 + 25sin^2t = 25
x^2 =25 - 25sin^2t
taking square root on both sides
x= √(25 - 25sin^2t)
= √25(1-sin^2t)
= 5√(1-sin^2t)
2)
Given rectangular equation:
y=-3x+4
Parametric equations x=2-t and y=?
Find parametric equation for y
Substituting value of x=2-t in given rectangular equation, we get
y= -3(2-t) +4
= -6+3t+4
= -2+3t !
write the equations of the line passing through the point (2,4) and parallel to the line whose equation is y=x
Answer:
y = x + 2
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = x is in this form
with slope m = 1 and c = 0
• Parallel lines have equal slopes, thus
y = x + c ← is the partial equation of the parallel line
To find c substitute (2, 4) into the partial equation
4 = 2 + c ⇒ c = 4 - 2 = 2
y = x + 2 ← equation of parallel line
The equation of the line parallel to y = x and passing through the point (2, 4) is y = x + 2.
Explanation:In mathematics, two lines are said to be parallel if they have the same slope. The given line is y = x, and because its slope is 1 (the coefficient of x), the line we're looking for, which is parallel to this, will also have a slope of 1.
To find the equation of a line with a given slope that passes through a certain point, we use the point-slope form of a linear equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of the point. Substituting 1 for m, and the given point (2, 4) into this equation gives us y - 4 = 1(x - 2), which simplifies to y = x + 2, the equation of the line we were seeking.
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Use a half-angle identity to find the exact value of sin75
[tex]\bf \textit{Half-Angle Identities} \\\\ sin\left(\cfrac{\theta}{2}\right)=\pm \sqrt{\cfrac{1-cos(\theta)}{2}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 75\cdot 2\implies 150\qquad \qquad \qquad \cfrac{150}{2}\implies 75 \\\\[-0.35em] ~\dotfill\\\\ sin\left( \cfrac{150^o}{2} \right)=\pm\sqrt{\cfrac{1-cos(150^o)}{2}}\implies sin\left( \cfrac{150^o}{2} \right)=\pm\sqrt{\cfrac{1-\left( -\frac{\sqrt{3}}{2} \right)}{2}}[/tex]
[tex]\bf sin\left( \cfrac{150^o}{2} \right)=\pm\sqrt{\cfrac{1+\frac{\sqrt{3}}{2}}{2}}\implies sin\left( \cfrac{150^o}{2} \right)=\pm\sqrt{\cfrac{\frac{2+\sqrt{3}}{2}}{2}} \\\\\\ sin\left( \cfrac{150^o}{2} \right)=\pm\sqrt{\cfrac{2+\sqrt{3}}{4}}\implies sin\left( \cfrac{150^o}{2} \right)=\pm\cfrac{\sqrt{2+\sqrt{3}}}{\sqrt{4}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill sin\left( \cfrac{150^o}{2} \right)=\pm\cfrac{\sqrt{2+\sqrt{3}}}{2}~\hfill[/tex]
The value of sin75 by using half angle identity is [tex]sin\dfrac{[150^0]}{[22]}=\pm\dfrac{\sqrt{2+\sqrt{3}}}{2}[/tex]
What is trigonometry?Trigonometry is the branch of mathematics that set up a relationship between the sides and angle of the right-angle triangles.
Value of sin75 by using half angle identity is calculated as:-
Half angle identity for is,
[tex]sin\dfrac{\theta}{2}=\pm\sqrt{\dfrac{1-cos\theta}{2}[/tex]
[tex]\dfrac{\theta}{2}=\dfrac{150}{2}[/tex]
[tex]sin\dfrac{150}{2}=\pm\sqrt{\dfrac{1-cos150}{2}[/tex]
[tex]sin \dfrac{150}{2}=\pm\sqrt{\dfrac{{1+\dfrac{\sqrt{3}}{2}}}{2}[/tex]
[tex]sin \dfrac{150}{2}=\pm\sqrt{\dfrac{2+\sqrt{3}}{4}[/tex]
[tex]sin\dfrac{[150^0]}{[22]}=\pm\dfrac{\sqrt{2+\sqrt{3}}}{2}[/tex]
Hence, the value of sin75 is [tex]sin\dfrac{[150^0]}{[22]}=\pm\dfrac{\sqrt{2+\sqrt{3}}}{2}[/tex].
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If f(x) and g(x)=x, determine the value(s) of x that satisfy the f(x)=g(x).
Answer: All values make a true statement.
Step-by-step explanation: Since f(x) and g(x)=x, f(x)=g(x) can also be written as x=x. That is a true statement so all values of x works.
Hope this helps!
The blue figure is rotated 90° counterclockwise around the origin and then reflected across the x-axis. In which quadrant will the transformed image be located?
The transformed image will be located in Quadrant III.
To solve this problem, we can think about what happens to the hour hand when it is rotated 90 degrees counterclockwise around the origin, and then reflected across the x-axis.
Rotation: When the hour hand is rotated 90 degrees counterclockwise around the origin, it moves to the position of the 3 o'clock hour. This means that the x and y coordinates of the tip of the hour hand are swapped. For example, if the tip of the hour hand was originally at (3, 4), after the rotation it would be at (4, 3).
Reflection: When the hour hand is reflected across the x-axis, its y-coordinate is multiplied by -1. This means that if the tip of the hour hand was at (4, 3) after the rotation, after the reflection it would be at (4, -3).
Now, let's look at the quadrants:
Quadrant I: x > 0, y > 0
Quadrant II: x < 0, y > 0
Quadrant III: x < 0, y < 0
Quadrant IV: x > 0, y < 0
Before the rotation, the tip of the hour hand is in Quadrant I, because both its x and y coordinates are positive.
After the rotation and reflection, the tip of the hour hand is in Quadrant IV, because its x-coordinate is still positive and its y-coordinate is now negative.
Therefore, the image will be in Quadrant III.
A triangle has side lengths of 13 2 and 13 what is the measure of the smallest angle?
Answer:
13
Step-by-step explanation:
Triangles consists of 180 degrees. 132 + 13 = 145 180 - 145 = 35, therefore the smallest measurement of the triangle is 13 since 132 and 35 are larger numbers.
If 1/2 of all the books at a bookstore are fiction and 3/5 of the books in fiction
are mysteries, what fraction of the store's books are fiction mysteries?
Answer:
[tex]\frac{3}{10}[/tex]
Step-by-step explanation:
we know that
To find the fraction of the store's books that are fiction mysteries, multiply 1/2 by 3/5
so
[tex](\frac{1}{2})(\frac{3}{5})=\frac{3}{10}[/tex]
The ages (in years) of eight passengers on a bus are listed below.
10 7 26 16 21 43 40 30
The measure of center is found to be 23.5 yr. Which measure of center is used?
mean
median
mode
midrange
Answer: median
Step-by-step explanation: Statistical Mathematics
First, we re-arrange the given data in ascending or descending order.Then we count the no. of data (here 8).If the data given is odd in count then the middle value of the arranged data is the median.but if the case is of even no. of data then we take the average of the two middle values of the arranged data.7, 10, 16, 21, 26, 30, 40, 43
Here we have an even number of the data entries so we get two values in the mid.
Now take the average of those mid values:
[tex]\frac{21+26}{2}[/tex] = 23.5Subtract the least value from the greatest value to find the range.For midrange add together the greatest value and the least value and divide by 2.Mean is the average of all the values.Mode is the value that occurs the most number of times.Which of these could be the graph of f(x) = In x + 2?
Answer:
C
Step-by-step explanation:
ln 1 = 0, so f(1) = 0 + 2 = 2. So we need to find a graph that passes through (1, 2). Only graph C fits.
The graph of the function f(x) = In x + 2 is the graph b
Which could be the graph of f(x) = In x + 2?The parent function for the natural log function is f(x) = ln(x).
To transform the graph of f(x) = ln(x) to the graph of f(x) = ln(x) + 2, we perform the following transformation:
Vertical Shift:
The graph is shifted two units upward.
This is because adding 2 to the output of the function has the same effect as shifting the graph two units upward.
So, the graph of f(x) = In x + 2 is b
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Solve the inequality statement:
4r – 5 < 5r+7
Answer:
r > - 12
Step-by-step explanation:
4r – 5 < 5r+7
4r - 5r < 7 + 5
-r <12
r > - 12 (when you multiply or divide a negative number, the sign will be flipped)
Answer:D
Step-by-step explanation:
can i have help this it is the picture below
Answer:
150 minutes, or 2 1/2 hours
Step-by-step explanation:
You want to know the time required to empty a pond with the given dimensions if the level drops 20 cm in the first 30 minutes of pumping.
Average depthThe average depth of the pond is the average of the depth at the shallow end and the depth at the deep end:
(0.6 m + 1.4 m)/2
= 2.0 m/2 = 1.0 m = 100 cm
If the level drops at the rate of 20 cm in 30 minutes, the time required for dropping the level by 100 cm is ...
t/(30 min) = (100 cm)/(20 cm)
t = 5·30 min = 150 min
The time to empty the pond completely is 150 minutes, or 2 1/2 hours.
__
Additional comment
As you know from the formula for the area of a trapezoid —
A = 1/2(b1 +b2)h
the area is equivalent to that of a rectangle with base equal to the average of the two base lengths.
Here, the volume of the pond is the product of the area of the trapezoidal face and the pond width (1 m). That trapezoidal face has an area equivalent to that of a rectangle 1 m high and 2 m wide. The 1 m height is the depth we used in the above calculation.
Cuantos días tiene el año?
Answer:
One year has 365 days
Step-by-step explanation:
The question in English is
How many days does the year have?
we know that
[tex]1\ year=12\ months[/tex]
[tex]1\ year=52\ weeks[/tex]
[tex]1\ year=365\ days[/tex]
therefore
One year has 365 days
Answer:
365
Step-by-step explanation:
What is the area of the square helicopter landing pad?
H
30 m
-
30 m-
30 x 30=900 square meters
The area of the square helicopter landing pad is equal to 900 square meters.
How to calculate the area of a square?In Mathematics and Euclidean Geometry, the area of a square can be calculated by using this formula;
A = [tex]s^2[/tex]
Where:
A is the area of a square.s is the side length of a square.By substituting the given side lengths into the formula for the area of square, we have the following;
Area of square, A = 30 × 30
Area of square, A = 900 square meters.
In this context, we can logically deduce that the area of the square helicopter landing pad is equal to 900 square meters.
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Plz help for financial algebra
Answer: $277.91
Step-by-step explanation:
The average of the 2 highest salaried quarters is:
[tex]\dfrac{\$13,500+\$12,775}{2}=\dfrac{\$26,275}{2}=\$3,137.50[/tex]
Divide that by 26 (weeks) to find out her average weekly salary:
[tex]\dfrac{\$3,137.50}{26}=\$505.29[/tex]
Multiply that by 55% (0.55) to calculate her weekly unemployment benefit:
[tex]\$505.29(0.55)=\$277.908[/tex]
Round to the nearest penny --> $277.91
Amy is 3 years older than her brother. Together, their ages are 14. Amy wrote the equation x+(x+3)=14 to represent the sum of their ages. What does x represent in Amy’s equation?
Amy’s age
Amy’s brother’s age
the sum of their ages
the difference of their ages
She is 3 years older than her brother and has (x+3) in the equation means that x is being used as her brothers age. (She is adding 3 to her brothers age)
The answer is Amy's brother's age.
Answer:
B
Step-by-step explanation:
fill in the missing exponents in each box & show reasoning (5^?)^3=5^1
the missing exponent would be 2
[tex]\bf (5^?)^{3}=5^1\implies 5^{?\cdot 3}=5^1\implies \stackrel{\textit{same base, the exponents must also be the same}}{?3=1\implies ?=\cfrac{1}{3}}[/tex]
Find the value of x in the following equation: x⁄2 + 2x⁄5 = 18.
Answer:
x = 20
Step-by-step explanation:
Solve for x by simplifying both sides of the equation, then isolating the variable.
let's take a peek, we have two fractions, let's get the LCD of all fractions, in this case the LCD of 2 and 5, hmmmm that'd be 10.
now, let's multiply both sides by the LCD of 10 to do away with the denominators and proceed.
[tex]\bf \cfrac{x}{2}+\cfrac{2x}{5}=18\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{10}}{10\left( \cfrac{x}{2}+\cfrac{2x}{5} \right)=10(18)}\implies 5x+4x=180 \\\\\\ 9x=180\implies x=\cfrac{180}{9}\implies x=20[/tex]
What is the function rule for the following situation? Rex paid $20 for a membership to the pool and pays $300 each time he goes to the pool
https://brainly.com/question/12669821
f(x)=20+300x
Also, Rex should find a different pool because that fee is ridiculous.
The function rule in the given situation is a linear function and can be represented as C=20+300x. In this rule, 'C' is the total cost in dollars, '20' is the initial membership fee, '300' is the cost of each pool visit and 'x' is the number of pool visits.
Explanation:The function rule in this situation is called a linear function. A function rule describes how to convert an input (in this case, the number of times Rex visits the pool) into an output (the total cost).
The function rule for this specific situation could be represented as C=20+300x, where 'C' is the total cost, '20' is the initial membership fee, '300' is the cost each time Rex goes to the couple, and 'x' is the number of times Rex goes to the pool.
So, each time Rex goes to the pool, you would multiply that number by 300 (the cost per visit), and then add 20 (the initial fee) to get the total cost.
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A sales clerk makes $259 in commission on a $170,000 worth car. What’s her commission rate?
Answer:
259 divided by 170,000 gives you the % 259 is out of 170,000 which is .0015. Move the decimal two spaces to the right to get the actual commission rate=.15%
Write the equation of the line that is perpendicular to the liney=-1x/ 3 + 2 and goes through the point (5, 6)
Answer:
y = 3x-9
Step-by-step explanation:
y = -1/3 x +2
The slope is -1/3
We want a line perpendicular to it
Perpendicular lines have negative reciprocal slopes
The new slope is -(-3/1) = 3
We have a slope m=3 and a point (5,6)
We will use point slope form to find the line
y-y-1 = m(x-x1)
y-6 = 3(x-5)
In slope intercept form
Distribute
y-6 = 3x-15
Add 6 to each side
y-6+6 = 3x-15+6
y = 3x-9
Find the surface area of the rectangular prism
Answer:
52 IN. FOR SURE
Step-by-step explanation:
Formula: A=2(wl+hl+hw)
Fill the formula and you'll understand I promise or search formula for surface area of a rectangular prism fill the numbers and your all set!!!
PLZZZ MARK BRAINLIST!!! PLZZ
Liliana wants to start a seventh-grade computer club at Hamden Middle School. She surveyed 20 seventh-grade students at the town park. She asked each student how many hours they spend on their computers each week. She obtained the following results.
8, 15, 0, 11, 12, 13, 16, 13, 0, 4, 17, 14, 30, 13, 5, 12, 1, 13, 12, 21
What is the ratio of the total number of students who used their computers to the total number of students surveyed?
Answer:
Ratio would be 9 : 10
Step-by-step explanation:
Liliana surveyed 20 students that how many hours they spend on their computers each week.
The results are as follows
8, 15, 0, 11, 12, 13, 16, 13, 0, 4, 17, 14, 30, 13, 5, 12, 1, 13, 12, 21
In the above results 2 students did not use computers out of 20 students.
Therefore, The total number of students who used their computers = 20 - 2 = 18
So the ratio would be 18 : 20 or 9 : 10
Answer:
C. 18/20 or 9/10
What is the value of v in the equation 3(2v + 1)=-15(5v + 16)
Answer:
v = -3
Step-by-step explanation:
Step 1: Use the Distributive Property
6v + 3= -75v - 240
Step 2: Isolate v
81v = -243
Step 3: Simplify
v = -3
The value of v in the equation 3(2v + 1)=-15(5v + 16) is -3 after rearranging and solving.
What is linear equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
We have a linear equation in one variable:
3(2v + 1)=-15(5v + 16)
6v + 3 = -75v -240
6v + 75v = -240 - 3
81v = -243
v = -3
Thus, the value of v in the equation 3(2v + 1)=-15(5v + 16) is -3 after rearranging and solving.
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The diagram shows the floor plan for Harry’s new tree house. The entry Terrance on the tree house is shaped like an isosceles trapezoid
The entry has an area of (blank) square feet the total area of the tree house is (blank) square feet
Answer:
The entry terrace has an area of
48
square feet. The total area of the tree house including the entrance, porch, and side deck is
350
square feet.
Step-by-step explanation:
I just took a test with this question and this was the right answer
~Please mark me as brainliest : )
find the area of the trapezoid?
Answer: 52.5 units squared
Step-by-step explanation:
The formula to find the area of a trapezoid is, (base1 + base2)/2 * height
So you find out which variable is which
Base 1 = 4
Base 2 = 3
Height = 15
Then plug it into the formula,
(4+3)/2 * 15
Then solve the order of operations
7/2 * 15
3.5 * 15
52.5 is the answer
Answer:
52.5
Step-by-step explanation:
Formula
Area = (1/2) (a + b)*h
Givens
a = 4
b = 3
h = 15
Solution
Area = (4 + 3)*15/2
Area = 7 * 15/2
Area = 105/2
Area = 52.5