For the parabola y = 2x² + 3x + 1,
a. i. The x-intercepts are x = -1/2 and x = -1
a. ii. The y-intercept is y = 1
b. i. The line of symmetry is at x = -3/4
b. ii. The equation of the line of symmetry is x = -3/4
c. The vertex is at (-3/4, -1/8)
What is a parabola?A parabola is part of a conic section which is a curve.
a. i. Find the x - interceptGiven the parabola y = 2x² + 3x + 1, we find the x- intercept when y = 0.
So, y = 2x² + 3x + 1
2x² + 3x + 1 = 0
2x² + 2x + x + 1 = 0
Factorizing, we have
2x(x + 1) + (x + 1) = 0
(2x + 1)(x + 1) = 0
2x + 1 = 0 or x + 1 = 0
2x = -1 or x = -1
x = -1/2 or x = -1
So, the x-intercepts are x = -1/2 and x = -1
ii Find the y intercept?Given the parabola y = 2x² + 3x + 1, we find the y- intercept when x = 0.
So, y = 2x² + 3x + 1
So, y = 2(0)² + 3(0) + 1
y = 0 + 0 + 1
y = 1
So, the y-intercept is y = 1
b. i. Where is the line of symmetry of this parábola?The line of symmetry passes through the vertex of the parabola where dy/dx = 0.
So, y = 2x² + 3x + 1
dy/dx = d(2x² + 3x + 1)/dx
= d(2x²)/dx + d3x/dx + d1/dx
= 4x + 3 + 0
= 4x + 3
So, when dy/dx = 0,we have
4x + 3 = 0
4x = -3
x = -3/4
So, the line of symmetry is at x = -3/4
ii. Write its equationThe equation of the line of symmetry is x = -3/4
c. Find the coordinates of the vertex?Since the vertex of the parabola is a t x = -3/4, substituting this into the equation, we have
y = 2x² + 3x + 1
y = 2(-3/4)² + 3(-3/4) + 1
y = 2(9/16) - 9/4 + 1
y = 9/8 - 9/4 + 1
y = (9 - 18 + 8)/8
y = (17 - 18)/8
y = -1/8
So, the vertex is at (-3/4, -1/8)
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5 = 3 + 2x
What is the answer?
Answer:
x = 1
Step-by-step explanation:
GivensWe are given an equation that will ultimately lead to a solution for the variable x.
To begin, we will first subtract 3 from both sides to move the constants to each side:
[tex]5=3+2x\\\\5-3=3-3+2x\\\\2=2x[/tex]
Then, divide both sides of the equation by 2 to remove the leading coefficient and isolate the variable:
[tex]2=2x\\\\\displaystyle \frac{2}{2}=\frac{2x}{2}\\\\x=1[/tex]
Final AnswerTherefore, x = 1.
Answer: x = 1
Step-by-step explanation: The equation is true without changing the value of x so x can only equal 1 so that the value of the numbers go unchanged. :)
An investment of $500 is made at 2.8% nominal yearly interest compounded quarterly. Write an equation that models the amount A the investment is worth t years after the principal has been invested
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightAlgebra IFunctions
Function NotationCompounded Interest Rate Formula:
[tex]\displaystyle A = P \bigg( 1 + \frac{r}{n} \bigg) ^t[/tex]
Let's compile what we know (what we are given).
An investment of $500 means that our principle amount P equals 500.
2.8% nominal yearly interest means that our rate r equals 0.028.
Compounded quarterly means that our compounded rate n equals 4.
So, we have:
[tex]\displaystyle\begin{aligned}P & = 500 \\r & = 0.028 \\n & = 4 \\\end{aligned}[/tex]
Step 2: WorkIn order to find an equation that models the final amount A in terms of t years, we can simply substitute in our known variables:
[Compounded Interest Rate Formula] Substitute in variables:∴ our equation that models the amount A the investment is worth t years after the principle has been invested is:
[tex]\displaystyle \boxed{ A = 500(1.007)^t }[/tex]
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___
Topic: Algebra I
Unit: Interest Rate Formulas
e altitude of a triangle is increasing at a rate of 1 cm!min while the area of the triangle is increasing at a rate of 2 cm !min. at what rate is the base of the triangle changing when the altitude is 10 cm and the area i
The base of the triangle is decreasing at the rate of 1.6 cm/min.
Given
dh/dt = 1 cm/min
dA/dt = 2 cm2/min
When A = 100 cm2, h = 10 cm
We know that
A = 1/2 × b × h
b = 20 cm
Let us differentiate both sides
dA/dt = 1/2 [b dh/dt + h db/dt]
Substituting the values
2 = 1/2 [20 × 1 + 10 × db/dt]
By further simplification
4 = 20 + 10 db/dt
db/dt = -1.6 cm/min
Therefore, the base of the triangle is decreasing at the rate of 1.6 cm/min.
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5(2x-3) < 35, the solution set is {x|x <(blank) }
Answer:
x<5
Step-by-step explanation:
5(2x-3) < 35
_____ ___
5 5
2x-3<7
2x<10
_ _
2x 2x
x < 5
Determine if the table shows a proportional relationship.
IT'S DUE TOMORROW
x 18.6 54.3 58.8
y 6.2 18.1 19.6
Yes, it is proportional because all y over x ratios are equivalent to one third.
Yes, it is proportional because all y over x ratios are equivalent to one fourth.
No, it is not proportional because 18.6 over 6.2 does not equal 18.1 over 54.3.
No, it is not proportional because 18.1 over 54.3 does not equal 58.8 over 19.6.
It is proportional because all y over x ratios are equivalent to one-third. I.e., the proportionality constant of the given table is k = 1/3.
How to determine the proportionality relationship?The proportionality relationship between two variables is determined by its proportionality constant. The proportionality constant is the ratio of y over x.
I.e., k = y1/x1 = y2/x2 = y3/x3 = ... = yn/xn
Calculation:The given data table is as follows:
x - 18.6, 54.3, 58.8
y - 6.2, 18.1, 19.6
So, the proportionality constant is calculated as
y1/x1 = 6.2/18.6 = 1/3
y2/x2 = 18.1/54.3 = 1/3
y3/x3 = 19.6/58.8 = 1/3
Since all three ratios are the same, then the data in the table are proportional to each other.
Therefore, option A is correct.
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What do I do here
I am really confused
Answer:
See below
Step-by-step explanation:
This is the slope formula. Do they give you two points? If they give you two points like (2,3 ) and (5,7) You subtract the y's over the subtraction of the x's
An ordered pair is in the form (x,y)
So in the example that I give the y's are 7 and 3 and the x's are 5 and 2
[tex]\frac{7-3}{5-2}[/tex] = [tex]\frac{4}{3}[/tex]
I am thinking that they gave you two points and they want you to find the slope.
Answer:
Step-by-step explanation:
Ok so this is how to find slope of a line that passes through 2 points.
(x1,y1) and (x2,y2)
and to find the slope of the line (m) you have to do (y2-y1)/(x2-x1)
you just have to plug in the points
19. In Juneau, Alaska, the temperature on Tuesday at noon was -20°F. The temperature then
decreased 1.25°F each hour for the next 5 hours.
In Tacoma, WA, the temperature on Tuesday at 5:30 pm was 2/3 warmer than the temperature
recorded in Juneau, Alaska at the same moment in time.
What is the temperature in Tacoma, WA? Show your thinking.
The temperature at Tacoma WA is -26 5/12°F.
What is the temperature?The temperature at 5pm in Juneau Alaska: temperature at noon + total change in temperature
-20 + (-1.25 x 5)
-20 - 6.25 = -26.25°F = -26 1/4°F
The temperature in Tacoma WA = termpature in Juneau + difference in temperature
-26 1/4°F - 2/3 = -26 5/12°F
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Which of the following values is not an irrational number?
1.73 repeating where the 3 is repeating, pi, the cube root of 90, and the square root of 60
1.73 repeating where the 3 is repeating
π
the cube root of 90
the square root of 60
All of the given numbers are irrational numbers.
What is an Irrational number:A real number that can't be stated as a ratio of integers is said to be an irrational number. In simple words, an Irrational number is a number that cannot be represented as p/q where q ≠ 0.
An irrational number's decimal expansion is neither ending nor recurrent i.e the decimal expansion of an irrational number will be a Non-terminating decimal number.
Here are some examples of Irrational Numbers
Here we have
[tex]1.7 \overline {3}[/tex] , π, ∛90, and √60
If we observe the given numbers
[tex]1.7 \overline {3}[/tex] = 1.73333333..... is a Non -terminating decimal number
π is an example of an irrational number
∛90 cannot be represented as a p/q form
√60 = √4 × 15 = 2√15 where √15 is an irrational number
From above information
The given numbers π, √90, and √60 all are irrational numbers
Therefore,
All of the given numbers are irrational numbers
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Which of the following equations represents a line that passes through the points (8,-14) and (-4,16)?
Answer: y=-15/6x+6
Step-by-step explanation:
Find slope using slope equations y2-y1/x2-x1
16-(-14)/-4-8
30/-12
slope equals to -15/6
y=-15/6x+b
-14=-15/6(8)+b
b=6
y=-15/6x+6
Select the correct answer from the drop-down menu. An energy drink company claims that its product increases students' memory levels. To support its claims, the company issues advertisements claiming that 8 out of 10 people (chosen randomly from across the country) who tried their product reported improved memory. The missing component in this study is a .
Here the missing component in this study is control group.
What is a control group in an experiment?A control group is a group of participants used to compare different groups in a test or experiment. Consists of participants who did not receive an experimental treatment. Some experiments use multiple control groups to collect more information and improve the accuracy of the results. For example, many medical studies include positive, negative, and placebo control groups to better understand the effectiveness of a treatment or drug.
Given the circumstances, the energy drink company claims that its products boost memory in students. To back up its claims, the company publishes an ad claiming that eight out of 10 people (randomly selected across the country) who tried the product reported improved memory.
But they didn't compose a control group to compare the results for test.
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The complete question is as follows:
An energy drink company claims that its product increases students' memory levels. To support its claims, the company issues advertisements claiming that 8 out of 10 people (chosen randomly from across the country) who tried their product reported improved memory. The missing component in this study is a
A) control group
B) sample group
C) population
D) test group
Answer:
A. Control Group
Plato
Lillian bought snacks for her team's practice. She bought a bag of apples for $2.36 and a 4-pack of juice bottles. The total cost before tax was $6.48. How much was each bottle of juice?
The price of each bottle of juice is $1.03
What is an equation?
A mathematical expression with an equals sign is referred to as an equation. Algebra is widely used in equations. When performing calculations but unsure of the precise amount, algebra is used.
Let 'x' be the price of each bottle of juice. Then,
According to the question,
2.36 + 4x = 6.48
or, 4x = 6.48 - 2.36
or, 4x = 4.12
or, x = 4.12/4
or, x = 1.03
Hence, the price of each bottle of juice is $1.03
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How do you write the equation for a line that passes through points (2,1) and (0,7)?
The equation for a line that passes through points (2,1) and (0,7) is y-7 = -3x or y= -3x+7
To find the equation of a line passing through two points you must first find the slope of the line.
The slope can be found by using the formula: [tex]m=\frac{y2-y1}{x2-x1}[/tex]
Where m is the slope and (x1,y1) and (x2,y2) are the two points. Substituting the two points given in the problem allows us to calculate the slope as:
[tex]m=\frac{7-1}{0-2} \\m= \frac{6}{-2} \\m= -3[/tex]
Next, we have to use the point-slope formula to find the equation for the line passing through these two points.
The point-slope formula states: (y-y1)=m(x-x1)
Where m is the slope and x1, and y1 are points from which the line passes.
Substituting the value of m and x1 and y1 in the given equation,
y-7=-3(x-0)
y-7= -3x-3*0
y-7+-3x
Solving for the slope-intercept form gives:
y-7+7= -3x+7
y-0 + -3x+7
y= -3x+7
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Martha loves to wear socks with crazy patterns. She finds a great deal for these kinds of socks at her favorite store, Rock Those Socks. There is a proportional relationship between the number of pairs of socks that Martha buys, x, and the total cost (in dollars), y.
Martha can buy 2 pairs of socks for $6. Write the equation for the relationship between x and y.
The equation which represent the proportional relationship between the number of pairs of socks that Martha buys, x, and the total cost (in dollars), y is y = bx
How to write equation?Number of pairs of socks that Martha buys = xTotal cost (in dollars) = yy = bx
Where,
b = constant
If y = $6 and x = 2 pairs
y = bx
6 = b × 2
6 = 2b
divide both sides by 2
b = 6/2
b = 3
In conclusion, the proportional relationship between number of pairs of socks that Martha buys, x, and the total cost (in dollars), y is y = bx
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If m∠ XYZ=68 and m∠ WXZ=71, find m∠ WZX
The measure of angle WZX is given as follows:
m < WZX = 41º.
How to obtain the angle measure?The measure of angle XYZ is of 68º, hence, due to the bisection, we have that:
m < WYZ = 34º.m < ZYX = 34º.The measure of angle WXZ is of 71º, hence the measure of angle WXY is found as follows:
m < 2 x 71 = 142º.
(as the bisection of angle X means that it is divided into two parts of equal measure).
As the figure is a parallelogram, the opposite angles have the same measure, thus the measure of angle WZY is given as follows:
m < WZY = 142º.
Then, applying the bisection, the measure of angle WZX is given as follows:
m < WZX = 0.5 x 142º = 41º.
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Equation for
You have c pieces of candy to share equally among 5 people. If
each person gets 32 pieces of candy, how much candy did you start
with?
Answer:
53 candy
Step-by-step explanation:
How do you convert 15 nm into km?
15 nm converted into km gives 1·5 × [tex]10^{-11}[/tex] km.
A process involving numerous stages, multiplication or division, especially a conversion factor, can be referred to as Unit Conversion. Additionally, rounding and determining the number of significant digits may be necessary steps in the procedure.
For measuring various properties, including length, weight, capacity, and temperature, many units of conversion are employed. We can change the units of measurement in mathematics to better understand.
Unit conversions are used to represent an amount in a different unit.
We have to convert 15 nm into km.
1 nm = [tex]10^{-12}[/tex] km
15 nm = 15 × [tex]10^{-12}[/tex] km
= 1·5 × [tex]10^{-11}[/tex] km
Thus, 15 nm converted into km gives 1·5 × [tex]10^{-11}[/tex] km.
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Which of the following pairs of linear equations are consistent or inconsistent XY 8 3x -- 3y 16?
The equations have no solutions and run parallel to one another. The pair of linear equations is hence inconsistent.
What are linear equations?Examples of linear equations include 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, and 3x - y + z = 3.
The definition of linear equations, the standard form for linear equations involving one variable, two variables, or three variables, and examples of each will all be covered in detail in this article.
Find the value of the variable that causes the equation to be true in order to solve linear equations.
Utilize the inverse of the variable's multiplier and divide or multiply both sides by it.
To obtain the variable value, simplify the result. Put your solution back into the equation to verify it.
So, we have the pair of linear equations:
x – y = 8 and 3x – 3y = 16
(a1/a2) = 1/3
(b1/b2) = -1/-3 = 1/3
(c1/c2) = 8/16 = 1/2
Since, we know that (a1/a2) = (b1/b2) ≠ (c1/c2).
The equations have no solutions and run parallel to one another. The pair of linear equations is hence incoherent.
Therefore, the equations have no solutions and run parallel to one another. The pair of linear equations is hence inconsistent.
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Complete question:
Find out whether the following pairs of linear equations x – y = 8 and 3x – 3y = 16 are consistent/inconsistent.
A recipe calls for 1.5 cups of flour to make 24 cookies. How many cookies can you make if you only have 1 cup of flour?
Answer:
16 cookies
Step-by-step explanation:
24 cookies/1.5 cups= 16 cookies per 1 cup
How many solutions can be found for the linear equation 4(2x+3)=2(4x+6)
Answer:
All real numbers are solutions
Step-by-step explanation:
Open the bracket
4(2x+3)=2(4x+6)
8x +12=8x +12
8X-8X=12- 12
=0
The equation is an identity hence, all real numbers are solutions(1,2,3,4 etc)
what is the quadratic equation in standard form with rational coefficents that has a root of 5+4i?
Answer:
x^2 -10x + 41 = 0
Step-by-step explanation:
They gave you a root. A root is a solution, is a zero, is an x-intercept(if real).
This is a working backwards problem. They gave you an answer and want the equation back again.
x = 5 + 4i
working backwards:
x - (5 + 4i) = 0
x - 5 - 4i = 0
This left side is a factor of your equation.
If 5+4i is a root then 5-4i is also a root. So do the same thing:
x = 5 - 4i
x - (5 - 4i) = 0
x - 5 + 4i = 0
x - 5 + 4i and
x - 5 - 4i are the factors. Multiply them together to find the quadratic equation. Use distributive property or difference of squares or an area model to model. See image.
Answer:
x² - 10x + 41 = 0
Step-by-step explanation:
complex roots occur as a conjugate pair.
given x = 5 + 4i , then x = 5 - 4i is also a root
the corresponding factors are
(x - (5 + 4i) ) , (x - (5 - 4i) ) , that is
(x - 5 - 4i) and (x -5 + 4i)
the quadratic equation is then the product of the factors, that is
(x - 5 - 4i)(x - 5 + 4i) = 0
multiply each term in the second factor by each term in the first factor.
x² - 5x + 4ix - 5x + 25 - 20i - 4ix + 20i - 16i² = 0 ← collect like terms
x² - 10x + 25 - 16i² = 0 [ note i² = - 1 ]
x² - 10x + 25 + 16 = 0
x² - 10x + 41 = 0 ← in standard form
What is the range of the function?
f(x)=3^x-1 -6
Enter your answer, in interval notation, in the box.
range:
The range of the function is (-6 , ∞)
What is function ?
An expression, rule, or law in mathematics that specifies the relationship between an independent variable and a dependent variable (the dependent variable).
An example of a rule is a function, which produces one output for a single input. Alex Federspiel is the picture's author. Y=X2 is an illustration of this. You only get one output for y if you enter anything for x. The fact that x is the input value leads us to say that y is a function of x.
Functions can be roughly categorized into four sorts. based on the element Numerous to one function, onto function, one to one and onto function, into function.
[tex]3^{x - 1} > 0[/tex] for x ∈ R
Subtracting both sides by 6
[tex]3^{x - 1} - 6 > 0 - 6[/tex]
[tex]3^{x - 1} - 6 > - 6[/tex]
f (x) > -6
Range = (-6 , ∞)
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Write an explicit
formula for an, the nth term of the sequence 2, 4, 8, ..
Answer:
formula: Tn=a+(n-1)d
Step-by-step explanation:
Tn=2+(n-1)2
A trapezoid has an area of 35 mm2. What is the height of the trapezoid?
The height of the trapezoid is 5mm.
What is a trapezoid?
A trapezoid is a quadrilateral in American and Canadian English that has at least one pair of parallel sides. It is referred to as a trapezium in British and other varieties of English.
In Euclidean geometry, a trapezoid is invariably a convex quadrilateral. The trapezoid's parallel sides are referred to as its bases. If the remaining two sides are parallel, the trapezoid is a parallelogram and has two pairs of bases; if not, they are referred to as the legs (or lateral sides). A scalene trapezoid is a trapezoid without equal-sized sides.
Given: Area of trapezoid = [tex]35mm^{2}[/tex]
We know, area of trapezoid = [tex]\frac{(a+b)}{2} .h[/tex]
⇒ [tex]\frac{(8+6)}{2} .h=35[/tex]
⇒ [tex]7h=35[/tex]
⇒ [tex]h=5mm[/tex]
Hence, the height of the trapezoid is 5mm.
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Doug bought a new car for $25,000. He estimates his car will depreciate, or lose value, at a rate of 20% per year. The value of his car is modeled by the equation V=P(1-r)^t, where V is the value of the car, P is the price he paid, r is the annual rate of depreciation, and t is the number of years he has owned the car. According to the model, what will be the approximate value of his car after 4 1/2 years
According to the equation model V=P(1-r)^t, the approximate value of Doug's car after 4¹/₂ years is $9,200.
What is the equation?The equation is a mathematical statement or model that shows that two mathematical expressions are equal.
In this situation, the mathematical expressions equal to each other are V, the car's depreciated value, and P(1-r)^t, which combines variables.
Cost of the new car = $25,000
Depreciation rate per annum = 20%
Value of the car = V
The price paid initially = P = $25,000
The annual rate of depreciation = r
Depreciation period = 4¹/₂ years
Equation model: V=P(1-r)^t
Depreciated value after 4 1/2 years, V = $25,000 (1 - 0.2)^4¹/₂
= $9,158.93
= $9,200
Thus, using the equation model, after 4¹/₂ years, the value of Doug's car has depreciated to $9,200.
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kevin opened a savings account with texas national bank. his account has an apr of 1.75% compounded quarterly. if kevin opens his account with $2500, how long will it take for the account to earn $7500?
It will take 62.91 years for Kevin to make the amount $2500 to $7500 .
In the question ,
it is given that ,
the amount that Kevin deposited (P) in bank = $2500
the interest rate (r) = 1.75% ,
the compound interest is quarterly ,
So , n is = 4 .
let the time taken to earn $7500 be = x years ;
Substituting the values in the Amount formula for the Compound Interest ,
Amount = P(1 + r/n)ⁿˣ
7500 = 2500(1 + 0.0175/4[tex])^{4x}[/tex]
Simplifying further ,
we get ,
3 = (1.004375[tex])^{4x}[/tex]
After applying log both the sides ,
we get ,
log(3) = 4x*log(1.004375)
x = log(3)/(4*log(1.004375))
Simplifying further ,
we have ,
x = 62.91 years .
Therefore , the time taken is = 62.91 years
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Put the following rational numbers in order from least to greatest.
70%, 7%, 17%, -3.7%, 0.3%
Answer:
-3.7%, 0.3%,7%,17%,70%
Step-by-step explanation:
if abc is 1 degree and 25' what is acd? give your answer in degrees and minutes
If abc is 1 degree and 25', the angle ACD is 178° 35'.
What is angle?An angle is a figure in Euclidean geometry formed by two rays, called the sides of the angle, that share a common endpoint, called the vertex of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are also formed by the intersection of two planes. These are termed dihedral angles. An angle is generated when two straight lines or rays meet at a shared terminus. The common point of contact is called the vertex of an angle.
Here,
∠ACB and ∠ACD are supplementary, so they add up to 180°.
First, convert to degrees (there are 60 minutes in a degree).
∠ACB = 1°25' = 1° + (25/60)° = 1.4167°
Now find the supplementary:
∠ACD = 180° − 1.4167°
∠ACD = 178.5833°
∠ACD = 178° 35'
The value of angle ACD is 178° 35' if abc is 1 degree and 25' .
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Select the values that make the inequalityt > -1 true.
(Numbers written in order from least to greatest going across.)
The values that make v ≥ 1 inequality true are 2, 3, 4, 5....
What is inequality in mathematics?
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions.
The inequality is given as:
v ≥ -1
The values of c for which the given inequality → v ≥ -1 is true will belong to the set (say A) -
A = [-1, - ∞)
Therefore, we have to select all numbers which fall in this set. The numbers are
-1, 0, 1, 2, 3,....
Rewrite properly as:
v ≥ 1
The above means that the true values of v are values greater than 1.
Hence, the values that make v ≥ 1 inequality true are -1, 0, 1 , 2, 3, 4, 5....
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Each minute, a faucet allows 4/7 gallons of water to enter a large tub, and the drain allows 5/7 gallons to leave the tub.
What is the change in the amount of liquid in the tub after 1/4 minute?
Enter your answer as a fraction in lowest terms in the box.
gal.
Answer: -1/28 gallon
Step-by-step explanation:
divide both numbers by 4 since its 1/4 a minute (-5/7 ÷ 4 and 4/7 ÷ 4) then add them, (-5/28 + 4/28= -1/28)
Answer: -1/28 gallon
Step-by-step explanation:
I can't decide in any way :')
Step-by-step explanation:
first thing that seems to be a given (and not mentioned in the description) : QM is a diameter (straight line) through the circle. otherwise this cannot be solved without further information.
remember, a half-circle has 180°.
so, arc angle MAP = 180 - 33 = 147°.
arc angle RMP = RM + MAP = 46 + 147 = 193°.
arc angle RSQ = 180 - 46 = 134°.
arc angle MRQ is the full half-circle = 180°.
arc angle MAP (as above) = 147°.
