Answer:
f⁻¹(x)= 9x + 18
Step-by-step explanation:
1. Replace f(x) with x, and x with y: x=1/9y-2
2. Solve for y:
x=1/9y-2
9x=y-18
9x+18=y
3. Replace y in step 2 with f⁻¹(x)
The vertex form of the equation of a horizontal parabola is given by x=1/4p(y-k)2+h , where (h, k) is the vertex of the parabola and the absolute value of p is the distance from the vertex to the focus, which is also the distance from the vertex to the directrix. You will use GeoGebra to create a horizontal parabola and write the vertex form of its equation. Open GeoGebra, and complete each step below.
Part A
Mark the focus of the parabola you are going to create at F(-5, 2). Draw a vertical line that is 8 units to the right of the focus. This line will be the directrix of your parabola. What is the equation of the line?
Part B
Construct the line that is perpendicular to the directrix and passes through the focus. This line will be the axis of symmetry of the parabola. What are the coordinates of the point of intersection, A, of the axis of symmetry and the directrix of the parabola?
Part C
Explain how you can locate the vertex, V, of the parabola with the given focus and directrix. Write the coordinates of the vertex.
Part D
Which way will the parabola open? Explain.
Part E
How can you find the value of p? Is the value of p for your parabola positive or negative? Explain.
Part F
What is the value of p for your parabola?
Part G
Based on your responses to parts C and E above, write the equation of the parabola in vertex form. Show your work.
Part H
Construct the parabola using the parabola tool in GeoGebra. Take a screenshot of your work, save it, and insert the image below.
Part I
Once you have constructed the parabola, use GeoGebra to display its equation. In the space below, rearrange the equation of the parabola shown in GeoGebra, and check whether it matches the equation in the vertex form that you wrote in part G. Show your work
Part J
To practice writing the equations of horizontal parabolas, write the equations of these two parabolas in vertex form:
focus at (4, 3), and directrix x = 2
focus at (2, -1), and directrix x = 8
Answer:
A. x = 3
B. A = (3, 2)
C. V = (-1, 2)
D. Left
E. See below.
F. p = -4
[tex]\textsf{G.} \quad x=-\dfrac{1}{16}(y-2)^2-1[/tex]
H. See attachment.
[tex]\textsf{J.} \quad x=\dfrac{1}{4}(y-3)^2+3[/tex]
[tex]x=-\dfrac{1}{12}(y+1)^2+5[/tex]
Step-by-step explanation:
Part AThe given focus of the parabola is F (-5, 2).
The equation for a vertical line is x = a.
Therefore, a vertical line that is 8 units to the right of the focus is:
[tex]\implies x=-5+8=3[/tex]
Therefore:
Directrix: x = 3Part BA line that is perpendicular to the directrix is a horizontal line.
The equation for a horizontal line is y = a.
As the horizontal line passes through the focus:
Axis of symmetry: y = 2The point of intersection, A, of the axis of symmetry and the directrix is:
(3, 2)Part CThe vertex of the parabola is the point that is halfway between the focus and the directrix.
Therefore:
Vertex = (-1, 2)Part DThe parabola will open sideways as the axis of symmetry is horizontal.
As a parabola never touches its directrix, this parabola will open to the left.
Part EThe absolute value of p is the distance between the vertex and the focus, and the distance between the vertex and the directrix.
For a sideways parabola, if p > 0 the parabola opens to the right, and if p < 0 the parabola opens to the left.
Part FSince the focus and directrix are 8 units apart, and the parabola opens to the left:
p = -4Part GVertex form of a sideways parabola:
[tex]\boxed{x=\dfrac{1}{4p}(y-k)^2+h}[/tex]
Given:
Vertex = (-1, 2) ⇒ h = -1, k = 2p = -4Substitute the values into the formula:
[tex]\implies x=-\dfrac{1}{16}(y-2)^2-1[/tex]
Part HSee attachment.
Part IRearrange the equation of the parabola shown in the attachment:
[tex]y^2+16x-4y=-20[/tex]
[tex]\implies 16x=-y^2+4y-20[/tex]
[tex]\implies 16x=-(y^2-4y+4)-16[/tex]
[tex]\implies 16x=-(y-2)^2-16[/tex]
[tex]\implies x=-\dfrac{1}{16}(y-2)^2-1[/tex]
This matches the equation in vertex form in part G.
Part JGiven:
Focus = (4, 3)Directrix: x = 2Therefore:
Vertex = ((4+2)/2, 3) = (3, 3)Parabola opens to the right, so p > 0.p = |4 - 3| = 1Therefore, the equation of the parabola with the given parameters is:
[tex]\boxed{x=\dfrac{1}{4}(y-3)^2+3}[/tex]
Given:
Focus = (2, -1)Directrix: x = 8Therefore:
Vertex = ((8+2)/2, -1) = (5, -1)Parabola opens to the left, so p < 0.p = -|2 - 5| = -3Therefore, the equation of the parabola with the given parameters is:
[tex]\boxed{x=-\dfrac{1}{12}(y+1)^2+5}[/tex]
Find the weight of a 176-cm male to the nearest kg whose BSA = 2.3.
The weight of the male that has a body surface area of 2.3 would be = 108kg
What is Body Surface Area (BSA)?The body surface area (BSA) is defined as the parameter that can be used to estimate the surface area of a person's body based on body weight and height.
The formula for BSA = √w×h/3600
The weight of the male = X
The height of the male = 176cm
The BSA = 2.3
From the formula, make w the subject of formula;
w = BSA²× 3600/h
w = 2.3²× 3600/176
w= 5.29×3600 /176
w = 19044/176
w = 108kg
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What number makes the expressions equivalent? Enter your answer in the box.
12 (–1.4m + 0.4) =
m + 0.2
The number makes the expression equivalent is m = 0.2584.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
Given, An expression 12(- 1.4m + 0.4) = m + 0.2.
- 16.8m + 4.8 = m + 0.2.
- 16.8m - m = 0.2 - 4.8
- 17.8m = - 4.6.
m = (-4.6/-17.8).
m = 0.2584.
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what did the mathematization do to practice over winter break
PLEASE HELP ASAP!!!! FULL ANSWER ONLY!!!
A person standing close to the edge on top of a 72-foot building throws a ball vertically upward. The quadratic function h = − 16 t^2 + 84 t + 72 models the ball's height above the ground, h , in feet, t seconds after it was thrown.
a) What is the maximum height of the ball? ____________ feet
b) How many seconds does it take until the ball hits the ground? ____________ seconds
a) The ball reaches a maximum height of 182.25 feet.
b) The ball has a travel time of 6 seconds.
How to find the main features of a free fall case by analyzing quadratic equations
In this case we find a kind of uniform accelerated motion known as free fall, in which a particle is accelerated uniformly by gravity and any effect of other forces are neglected. The height of the ball in terms of time is described by a quadratic equation, whose definition is shown below:
h(t) = h' + v' · t + 0.5 · g · t²
Where:
h' - Initial height, in feet.v' - Initial velocity, in feet per second.t - Time, in seconds.g - Gravitational acceleration, in feet per square second.First, write the complete the equation of the height of the ball:
h(t) = - 16 · t² + 84 · t + 72
a) Second, clear the expression on one side and use the discriminant of the quadratic formula:
- 16 · t² + 84 · t + [72 - h(t)] = 0
84² - 4 · (- 16) · [72 - h(t)] = 0
7056 + 64 · [72 - h(t)] = 0
7056 + 4608 - 64 · h(t) = 0
11664 - 64 · h(t) = 0
h(t) = 729 / 4 ft
h(t) = 182.25 ft
The maximum height of the ball is 182.25 feet.
b) Third, use the quadratic formula to determine the roots of the quadratic equation:
t = - 84 / [2 · (- 16)] ± [1 / [2 · (- 16)]] · √[84² - 4 · (- 16) · 72]
t = 2.625 ± (1 / 32) · 108
t₁ = 6 s and t₂ = -0.75 s
The travel time of the ball is 6 seconds.
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After reading a book for English class,100 students were asked whether or not they enjoyed it.
The number of students who like the book is 64 students.
How many students like the book?A fraction is simply a piece of a whole. The number is represented mathematically as a quotient where the numerator and denominator are split.
Given: Total students in English class = 100
Nine twenty-fifths of the class did not like the book.
Thus, the number of students did not like the book = 9/25 × 100
= 36
Therefore, the number of students like the book =Total students- students did not like the book will be:
= 100-36
= 64 students.
The number of students is 64.
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Complete question
After reading a book for English class, 100 students were asked whether or not they enjoyed it. Nine twenty-fifths of the class did not like the book. How many students liked the book?
In 2006 the median yearly family income was about 48,200 per year suppose the average annual rate of change since then is $1240 per year. Write and graph the equality for the annual family incomes y that are less than the median x years after 2006
The inequality for the annual family income is:
y < 1240x + 48,200
What is inequality?
An inequality in mathematics depicts the connection between two values in an algebraic statement that are not equal. One of the two variables on the two sides of the inequality may be greater than, greater than or equal to, less than, or less than or equal to another value, according to inequality signals.
Let x=0 be the year 2006
The inequality is:
y < 1240x + 48,200
Graphing:
Consider equal sign in place of < symbol
Take random values for x, find corresponding y values and plot them
Mark it as dashed line as shown in the graph because of the inequality symbol "< "
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Analyze and Persevere Last month, a truck driver made
5 round-trips to Los Angeles and some round-trips to San Diego.
Write an expression that shows how many miles he drove in all.
Identify and describe the part of the expression that shows how many miles he drove and trips he made to San Diego.
The expression that shows the number of miles he drove and trips he made to San Diego is; 3850 + yb
How to solve Algebra word problems?Let a round trip to Los Angeles be denoted by the variable; x
Let the number of miles in a round trip to Los Angeles be denoted by the variable a
Let a round trip to San Diego be denoted by the variable: y
Let the number of miles in a round trip to San Diego be denoted with the variable: b
This means that our algebraic equation representing the number of miles driven in all to be: xa + yb
Since the truck driver made 5 round-trips to Los Angeles, then it means that, we have the equation;
5a + yb as the total miles driven in all.
However, from the table attached, a round trip to Los Angeles is 770 miles. Thus;
Total miles driven = 5(770) + yb = 3850 + yb
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Aiden runs a farm stand that sells apples and strawberries. Each pound of apples sells
for $2 and each pound of strawberries sells for $3. Aiden made $80 from selling a
total of 35 pounds of apples and strawberries. Write a system of equations that could
be used to determine the number of pounds of apples sold and the number of pounds
of strawberries sold. Define the variables that you use to write the system.
Answer:
Step-by-step explanation:
Work
Find the radius of a square with perimeter 32cm
Answer: 5.66
Step-by-step explanation:
The perimeter of the square being 32, the sides are 8 units. The diameter of the circumscribing circle is the diagonal of the square which is 8 sec 45 = 11.3 units. Hence, the radius of the circle = 5.66 units.
Can someone help pls it’s due in about 2 hour
Answer:
see below
Step-by-step explanation:
A
The hypotenuse is ALWAYS the longest side of a right triangle, and located Directly Across from the Right Angle: c = √79
Pythagorean Theorem: a² + b² = c²
a = 2√5
b = ?
c = √79
(2√5)² + b² = (√79)²
20 + b² = 79
b² = 59
√b² = √59
b = √59 = KJ
B
tan=opposite side / adjacent side
From I, KJ is opposite; and IK is adjacent
tan(I) = √59/(2√5) ≈ 1.72
C
cos=adjacent side / hypotenuse
From J, KJ is adjacent; and IJ is the hypotenuse
cos(J) = √59/√79 ≈ 0.86
The length of a rectangle is 4 inches longer than it is wide. If the area is 32 square inches, what are the dimensions of the rectangle?
Answer:
width = 4in
length = 8in
Step-by-step explanation:
A = LxW = 32
L = W + 4
W(W+4) = 32
W² + 4W - 32 = 0
(W + 8)(W - 4) = 0
W ≠ -8 (dimensions cannot be negative)
W = 4in
L = W+4 = 4+4 = 8in
(3×10³) (2× 10²)
Write your answer in scientific notation
pleaseeee help lol, will mark brainliest
Transforming the function f(x) = 2x by compression of a factor of 1/4 gives g(x) = 1/2x
Transformation of a FunctionTransformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f(x) = x2 + 3 is obtained by just moving the graph of g(x) = x2 by 3 units up. Function transformations are very helpful in graphing the functions just by moving/expanding/compressing/reflecting the curve without actually needing to graph it from scratch.
In the given function;
f(x)= 2x;
After a compression by a factor of 1/4, the function becomes
g(x) = 2x * 1/4 = 1/2x
The compressed function g(x) is 1/2x
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Given f(x)=2x-2, find f(-3)
PLEASE HELP!!!!
Answer:4
Step-by-step explanation:
replace x with -3
f(x) = 2x-2
find f(-3)
f(-3) = 2(-3) - 2
f(-3) = -6 -2
f(-3) = -8
Which expression is equivalent to (7x3)?(x8)2?
A
14x10
B
49x10
C
14x'
D 49x’
The expression equivalent to 7x³ × (x⁸) × 2 is 14x¹¹.
How to find an equivalent expression?Two expressions are said to be equivalent if they have the same value irrespective of the value of the variable(s) in them.
In other words, two algebraic expressions are said to be equivalent if their values obtained by substituting the values of the variables are the same.
Therefore,
7x³ × (x⁸) × 2
Using the law of indices,
[tex](x^{a}) (x^{b} ) = x^{a+b}[/tex]
Hence,
7x³ × (x⁸) × 2 = 7x³ × 2x⁸
7x³ × 2x⁸ = 14x¹¹
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Perform The operations!
Step-by-step explanation:
Step 1: 2x²--7x²=9x²
Step 2: 2x--7x=9x
Step 3: -7-+2=-9
Overall answer: 9x²+9x-9
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Dan went for a drive in his new car. He drove at a speed of 64 miles per hour for 102.4 miles. For how many hours did he drive?
___ hours
Answer:
Step-by-step explanation:
Dan drove for 1.6 hours.
64 miles per hour (mph) is the unit rate so divide by 102.4 by 64 and you get 1.6
The number of men and women receiving bachelor's degrees each year has been steadily increasing. For the years 1970 through the projection of 2014, the number of men receiving degrees (in thousands) is given by the equation y=3.9x+441 , and for women, the equation is y=14.3x+312, where x is the number of years after 1970
The point (12,489) will be the solution of our given equations.
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given:
number of men given by y=3.9x+441 ,
and for women equation is y=14.3x+312,
Now, Let us find the coordinates of the point of intersection of both lines by equating our both equations as:
3.9x+ 441 = 14.3x + 312
14.3x - 3.9x = 441 - 312
10.4x= 129
x= 129/10.4
x = 12.40
and, y= 3.9(12.4)+ 441
y= 489.36
Hence, the point (12,489) will be the solution of our given equations.
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CAN SOMEONE HELP WITH THIS QUESTION?✨
Answer:
b = 40 , c = 70
Step-by-step explanation:
given a quadratic function in standard form
y = ax² + bx + c ( a ≠ 0 )
then the x- coordinate of the vertex is
x = - [tex]\frac{b}{2a}[/tex]
y = 5x² + bx + c ← is in standard form
with a = 5 and b = b , then
x = - [tex]\frac{b}{10}[/tex]
equating to the given x- coordinate of the vertex , that is - 4
- [tex]\frac{b}{10}[/tex] = - 4 ( multiply both sides by - 10 )
b = 40
then
y = 5x² + 40x + c
substitute (- 4, - 10 ) into the equation and solve for c
- 10 = 5(- 4)² + 40(- 4) + c
- 10 = 5(16) - 160 + c
- 10 = 80 - 160 + c
- 10 = - 80+ c ( add 80 to both sides )
70 = c
then b = 40 and c = 70
Two friends are renting a property with a monthly rent of $975.00. The total square footage is 1,225 square feet, the first bedroom is 150 square feet, and the second bedroom is 130 square feet. Determine how much the friend with the smaller bedroom will pay for rent if they split the cost based on the square footage of each bedroom. (2 points)
$479.50
$481.35
$495.50
$501.35
Answer:
(a) $479.50
Step-by-step explanation:
You want to know how much the friend with the smaller bedroom will pay for rent if they split the $975 cost based on the square footage of each bedroom. The two bedrooms in the 1225 square foot property are 130 and 150 square feet.
SplitApparently, the cost of the non-bedroom area is to be split evenly. That common area is ...
1225 ft² -(150 +130) ft² = 945 ft²
Then the proportion due from the occupant of the smaller bedroom is ...
(945/2 +130)/1225 = 241/490 ≈ 0.49184
RentThat fraction of the rent is ...
0.49184 · $975 = $479.54
The nearest answer choice is $479.50.
__
Additional comment
If we were to assume the split is based only on bedroom size, then the rent paid for the 130 ft² space would be (13/28)($975) ≈ $452.68. This would be equivalent to splitting the rent for the common area in the same proportion as for the bedrooms.
We note that half the rent is $487.50, a number less than either of the last two answer choices. Those can be rejected for that reason. (The smaller bedroom is expected to pay less than half of the rent.)
What is the image of the point (-6, 2) after a rotation of 180° counterclockwise
about the origin?
Answer:
[tex](6,2)[/tex]
Step-by-step explanation:
When rotating 180° about the origin, [tex](x,y) \longrightarrow (-x, -y)[/tex].
6-4x=6x-8=2
PLEASE HELP
The value of x is 1.2
What is the solution to an equation?
In order to make the equation's equality true, the unknown variables must be given values as a solution. In other words, the definition of a solution is a value or set of values (one for each unknown) that, when used as a replacement for the unknowns, transforms the equation into equality.
6-4x = 6x-8+2
group the variable and constant terms
6 +8 -2 = 6x +4x = 10x
12= 10x
=> x= 12/10 = 1.2
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Given that m_G = 81°, what is m
Answer:
180-81=
Step-by-step explanation:
Find an expression which represents the sum of (7x-y) and (5x+3y) in simplest terms.
An expression that represent the sum of (7x-y) and (5x+3y) in simplest terms is 12x + 2y
How to find the sum of the expressionsThe given expressions are (7x-y) and (5x+3y)
The applying the addition property of equality we have
= (7x - y) + (5x + 3y)
= 7x - y + 5x + 3y
collecting like terms
= 7x + 5x - y + 3y
= 12x + 2y (in simplest terms)
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[tex]5^{2x+1} =25[/tex]
Answer:x=1/2
Step-by-step explanation:
[tex]5^{2x+1} =5^{2}[/tex]
2x+1=2
2x=1
x=1/2
Equation with one solution 8x-3x+2-x=
[tex]\fbox{4x+2}[/tex]
Simplify:
[tex]8x-3x+2[/tex]
[tex]8x+-3x+2+-x[/tex]
Combine Like Terms:
[tex](8x+-3x+-x)+(2)[/tex]
[tex]4x+2[/tex]
What is the second term in the binomial expansion of (2r – 3s)12?
–73,728r11s
73,728r11s
608,256r10s2
–608,256r10s2
The second term in the binomial expansion of (2r - 3s)¹² is -73728 r¹¹ s option (A) is correct.
What is an expression?It is defined as the combination of constants and variables with mathematical operators.
It is given that:
(2r - 3s)¹²
Using nth term of the binomial expansion is:
= C (12, 2-1) (2r)¹²⁻²⁺¹ (-3s)²⁻¹
After solving
= -73728 r¹¹ s
Thus, the second term in the binomial expansion of (2r - 3s)¹² is -73728 r¹¹ s option (A) is correct.
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If Karla has 3 more nickels than quarters and they have a combined value of 135 cents, how many of each coin does she have?
Answer:
Step-by-step explanation:
n=3+d
n -d = 3
5n -5d = 15
5n +10d = 135
5n - 5d = 15
15d = 120
d= 8 dimes
n = 11 nickels
5x11 + 8x10 = 55 + 80= 135
What is the difference between the place value of a whole number and a place value of a decimal?
The main differences between the place value of the whole number and the decimal is given below
What is decimal ?
Integer and non-integer numbers are often expressed using the decimal numeral system. The Hindu-Arabic numeral system has been expanded to include non-integer values. Decimal notation is the term used to describe how numbers are represented using the decimal system.
A decimal is a number with a whole and a fractional component. Decimal numbers, which are in between integers, are used to express the numerical value of full and partially whole amounts.
Decimals can be classified as non-recurring (as in not repeating or ending), recurring (as in repeating or not terminating), and decimal fractions (a.k.a. converting a decimal into a fraction whose denominator is a power of ten).
The whole number part of the decimal number is the part that lies strictly to the left of the decimal point, and the place value of each digit in the whole number part is given .
The fractional part of the decimal number is the part that lies strictly to the right of the decimal point.
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