The range of the given function, f(x) = x^2 - 4x + 1, is {y: y ≥ -3}. This is determined by completing the square to find the vertex form of the parabola, which indicates that the minimum y-value is -3 (the y-coordinate of the vertex) and the function values extend to infinity.
Explanation:The question asks us to find the range of the function f(x) = x^2 - 4x + 1. In mathematics, the range of a function is the set of output values (the y-values) that the function produces when we substitute the domain (the x-values) into the function.
To determine the range of a quadratic function like this, we could complete the square for the function and adjust it to the vertex form of a parabola, which is f(x) = a(x - h)^2 + k, where k is the minimum or maximum y-value of the parabola.
For the function f(x) = x^2 - 4x + 1, the completed square form is f(x) = (x - 2)^2 - 3. So, the vertex of the parabola is (2, -3), and since the coefficient of (x - 2)^2 is positive, the parabola opens upwards. This means that the minimum y-value is -3 and the maximum y-value is infinity. So the range of the function is {y: y ≥ -3}.
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Mr. Pham wrote the equation below on the board. 18 - 7x = -20.5 What is the value of x?
Answer: The answer should be D(x=5 1/2)
Step-by-step explanation:
Trust me I did the equation and work.
Jim has three times as many comic books as Charles Charles has two thirds as many as Bob Bob has 27 books how many comic books does Jim have
Charles has 2/3 of bob
bob has 27
so 2/3*27 = 27 *2 = 54/3 = 18
Charles has 18
Jim has 3 times as many as Charles so he has 3 x 18 = 54
jim has 54 comic books
The area of a rectangle is 35 feet^2, and the length of the rectangle is 8 feet less than three times the width. Find the dimensions of the rectangle
The width of the rectangle is 5 feet, and the length is 7 feet.
To find the dimensions of the rectangle, set up the equation (3W - 8)W = 35, where W is the width and solve for W. You'll find that the width is 5 feet, and the length, which is 8 less than three times the width, is therefore 7 feet.
To find the dimensions of the rectangle with an area of 35 feet2 where the length is 8 feet less than three times the width, let's denote the width as W and the length as L. The problem gives us two equations:
Area: L imes W = 35Length Relationship: L = 3W - 8Now, replace L in the first equation with 3W - 8, so we have:
(3W - 8)W = 35
Expanding the equation, we get:
3W2 - 8W = 35
Solve this quadratic equation to find the width W, and then substitute W back into L = 3W - 8 to find the length L. Let's solve the quadratic equation:
3W2 - 8W - 35 = 0
By factoring or using the quadratic formula, we can find that W=5 and L=7 are the solutions that makes sense for the dimensions of the rectangle.
So, the width of the rectangle is 5 feet, and the length is 7 feet.
a.Find the perimeter of the triangle.
b.Find the area of the triangle.
c. If tile cost $6 per square centimeter, how much will it cost to tile the triangle?
the sum of three number 20. the sum of twice the first number, and 5 times the third number is 71. the difference between 4 times the first number and the scond number is 27. find the three numbers. Helllp
Car mart pays $110,000 rent each year for its two-story building. the space in this building is occupied by five departments as specified here. paint department 1,575 square feet of first-floor space engine department 2,925 square feet of first-floor space window department 1,845 square feet of second-floor space electrical department 765 square feet of second-floor space accessory department 1,890 square feet of second-floor space the company allocates 70% of total rent expense to the first floor and 30% to the second floor, and then allocates rent expense for each floor to the departments occupying that floor on the basis of space occupied. determine the rent expense to be allocated to each department.
First floor:
paint department 1,575 square feet
space engine department 2,925 square feet
Total = 4,500 square feet
Second floor:
space window department 1,845 square feet
electrical department 765 square feet
accessory department 1,890 square feet
Total = 4,500 square feet
Since 70% is allocated to the 1st floor, therefore the floor is receiving:
First floor expense = $110,000 * 0.70 = $77,000
Second floor expense = $110,000 - $77,000 = $33,000
The expense for each department is the proportion on the total expense per floor.
First floor:
paint department = (1575/4500) * 77,000 = $26,950
space engine department = 77,000 – 26,950 = $50,050
Second floor:
space window department = (1845/4500) * 33,000 = $13,530
electrical department = (765/4500) * 33,000 = $5,610
accessory department = 33,000 – 13530 – 5610 = $13,860
If prices Increase at a monthly rate of 1.4
%, by what percentage do they increase in a year?
An isosceles triangle with angles b and c having the same measure is shown. Find the measure of each angle whose degree measure is represented with variables. A= x+ 7y+41,B= 2y+13,C=6x+15
Answer:
A = 114°, B = C = 33°
Step-by-step explanation:
The triangle relationships let you write two equations:
A+B+C = 180
B=C
Substituting the expressions for A, B, and C, you have ...
(x+7y+41) +(2y+13) +(6x+15) = 180
7x +9y +69 = 180
7x +9y = 111
And the second equation gives ...
(2y+13) = (6x+15)
6x -2y =-2
3x -y = -1
Now, we can add 9 times this second equation to the first to eliminate the y-variable.
(7x +9y) +9(3x -y) = (111) +9(-1)
34x = 102
x = 3
Then the angle measures are ..
B = C = 6·3+15 = 33
A = 180 -2·33 = 114
The angles in the triangle are (A, B, C) = (114°, 33°, 33°).
The school choir has 84 members. The ratio of girls to boys in the color is 3: 4. How many members are girls?
show steps !
(05.02 MC)
Trevor solved the system of equations below. What mistake did he make in his work?
2x + y = 5
x − 2y = 10
y = 5 − 2x
x − 2(5 − 2x) = 10
x − 10 + 4x = 10
5x − 10 = 10
5x = 0
x = 0
2(0) + y = 5
y = 5
His work should have looked like this.
2x + y = 5
x − 2y = 10
y = 5 − 2x
x − 2(5 − 2x) = 10
x − 10 + 4x = 10
5x − 10 = 10 When he added in both sides, he subtracted 10 from 10,
5x = 20 instead of adding 10 from 10 to make 20.
x = 4
2(4) + y = 5
8 + y = 5
y = -3
Angle θ is in standard position. If (8, -15) is on the terminal ray of angle θ, find the values of the trigonometric functions.
Answer with explanation:
It is given that, Angle θ is in standard position.
A line from origin O to point , P(8,-15) is joined and then perpendicular to x and y axis, is drawn cutting X axis at Point M and Y axis at point N.
OM= 8 units
ON=P M=15 units
By Pythagorean Theorem
[tex]OM^2 + PM^2=OP^2\\\\ 8^2 +15^2=OP^2\\\\ OP^2=64 +225\\\\OP^2=289\\\\OP^2=17^2\\\\OP=17\\\\ Sin (\theta)=\frac{\text{Perpendicular}}{\text{Hypotenuse}}=\frac{-15}{17}\\\\Cos(\theta)=\frac{\text{Base}}{Hypotenuse}=\frac{8}{17}\\\\Tan(\theta)=\frac{\text{Perpendicular}}{Base}=\frac{-15}{8}\\\\ Cosec(\theta)=\frac{1}{Sin(\theta)}=\frac{-17}{15}\\\\ Sec(\theta)=\frac{1}{Cos(\theta)}=\frac{17}{8}\\\\ Cot (\theta)=\frac{1}{Tan(\theta)}=\frac{-8}{15}[/tex]
Point(8,-15), lies in Quadrant four. In Quadrant four Cosine and Secant Function are positive and all other trigonometric functions, Sine,Cosecant, Tangent,and Cotangent are Negative.
For which function is f(x) not equal to f-1(x)?
A- f(x)=2-x
B-f(x)=2/x
C=f(x)=-x
D=f(x)=2x
Answer:
[tex]f(x)= 2x[/tex]
Step-by-step explanation:
For which function is f(x) not equal to f^-1(x)
LEts check with each function, we find out inverse for each option
[tex]f(x)= 2-x[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]y= 2-x[/tex]
[tex]x= 2-y[/tex], Add y on both sides
[tex]x+y= 2[/tex], Subtract x from both sides
[tex]y= 2-x[/tex], Inverse is equal to f(x)
[tex]-f(x)= \frac{2}{x}[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]-y= \frac{2}{x}[/tex]
[tex]-x= \frac{2}{y}[/tex], cross multiply it
[tex]-xy = 2[/tex], divide by x on both sides
[tex]-y= \frac{2}{x}[/tex], Inverse is equal to f(x)
[tex]f(x)= -x[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]y= -x[/tex]
[tex]x=-y[/tex], Add y on both sides
[tex]x+y= 0[/tex], Subtract x from both sides
[tex]y=-x[/tex], Inverse is equal to f(x)
[tex]f(x)= 2x[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]y= 2x[/tex]
[tex]x= 2y[/tex], divide by 2 on both sides
[tex]\frac{x}{2}=y[/tex], Inverse is not equal to f(x)
The waitress and hostess at a restaurant share the tips at the end of every shift. Between the two of them, they earn an average of $75 in tips. The waitress take $55 and the hostess takes $20. If they continue to earn money at this rate, how much will the waitress receive if they earn $300 at the end of their next shift?
thy get 55/75 and you want to know x/300
55/75 x X/300
cross multiply 55 x 300 = 16500
divide that by 75
16500/75 = 220
x = 220
they will get $220
4% of what number is 6
4% = 0.04
divide 6 by 0.04 = 150
6 is 4% of 150
double check 150*0.04 = 6
Josh estimates the height of his desk. Which is the best estimate?
12 pounds of apples. Each pounds costs $3. If she gives the cashier two $20 bills, how much change should she receive.
Samantha loves to download new music. She originally had 52 songs but plans to purchase 2 new songs each week. She wants to know how many songs she will have after 93 weeks. Which equation should she use
there are 24 people using the gym, tha ratio of men to women is 2;1, hoiw many men are using the gym
2 men for every 1 woman
24/3 =8
8*2 = 16
there are 16 men and 8 women
Is 1 fifth of 186 the same as 25% of 186
1/5 = 0.20
25% = 0.25
therefore 1/5 is not the same as 25%
"4. Suppose y varies directly with x. Write a direct variation equation that relates x and y. Then find the value of y when x= 12. y = -10 when x = 2 "
Answer:
5/ 2 x
Step-by-step explanation:
There are 147 peaple attending a diner party if each table can seat 7 peaple how many tables are needed for the dinner party?
how to say 119,000,003 in two other ways?
You want to have $25,000 saved 6 years from now to buy a house. how much less do you have to deposit today to reach this goal if you can earn 5.5 percent rather than 5 percent on your savings? today's deposit is the only deposit you will make to this savings account this problem requires solving for
By moving from a 5% interest rate to a 5.5% interest rate, you would need to deposit $372.84 less today for your savings goal of $25,000 in 6 years.
Explanation:When planning for a savings goal, we use the formula for the future value of a single amount which is FV = PV * [tex](1 + r)^n[/tex], where PV is the present value or the amount you need to deposit today, FV is the future value or the goal of $25,000, r is the interest rate, and n is the number of years.
First, let's calculate how much you need to deposit today with an interest rate of 5% (r = 0.05), the formula will be rearranged to solve for PV which is PV = FV / [tex](1 + r)^n[/tex]:
PV = 25000 / [tex](1 + 0.05)^6[/tex] = $18724.08
Next, calculate the deposit for an interest rate of 5.5% (r = 0.055) using the same formula:
PV = 25000 / [tex](1 + 0.055)^6[/tex] = $18351.24
The difference between these two amounts is $18724.08 - $18351.24 = $372.84.
So, with an interest rate of 5.5% rather than 5%, you would need to deposit $372.84 less today to reach your goal of $25,000 in 6 years.
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Anyone Know This Answer ASAP ?
multiply 35 by tan(42)
this equals 31.5141
rounded to 31.51
Which is a ppssible number of distinct real roots for a cubic function select all that apply.
0
1
3
4
Try graphing y=x^3. It crosses the x-axis at (0,0), and this point represents the one and only real root.
Every form of a cubic function has a graph that crosses the x-axis in 1 or 3 places.
Thus, the correct answers to this particular problem are B and C.
Additionally, certain cubic function forms have graphs that cross the x-axis in one unique place, but which touch (but do not cross) the x-axis. Here you have one unique real root plus one repeated (duplicated) real root, for a total of 3 roots.
Using these facts, decide which of the four given answers are correct.
The parent function, f(x) = 5x, has been vertically compressed by a factor of one-half, shifted to the left three units and up two units.
The parent function, f(x) = 5x, has been vertically compressed by a factor of one-half, shifted to the left three units and up two units.
Explanation:To vertically compress the parent function f(x) = 5x by a factor of one-half, we multiply the function by the compression factor, which is 1/2. So the compressed function is f(x) = (1/2)(5x) = 2.5x.
To shift the function to the left three units, we subtract the shift amount from the x-variable. So the shifted function is f(x + 3) = 2.5(x + 3) = 2.5x + 7.5.
To shift the function up two units, we add the shift amount to the y-variable. So the final transformed function is f(x + 3) + 2 = 2.5x + 7.5 + 2 = 2.5x + 9.5.
1. Describe how factoring a quadratic expression ax2 + bx + c, where a ≠ 1, is different from factoring x2 + bx + c.
2. Two students factored 2x2 + 6x – 20. Keiko said that the factorization was (2x – 4)(x + 5). Ray gave the factorization as (x – 2)(2x + 10). Confirm that both of these factorizations are correct. Then explain why they are not complete.
3. Explain the relationship between the factors of a quadratic expression, the roots of the related quadratic equation, and the x-intercepts of the graph of the related function.
Answer:
1. Dividing the expressions [tex]ax^2+bx+c[/tex] by a is a different step.
2. Yes, both of these factorization are correct. They are not complete because they can be factored further.
3. The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.
Step-by-step explanation:
1.
To factorize the quadratic expressions [tex]ax^2+bx+c[/tex] first we divide it by a. Then we factorize it same as [tex]x^2+bx+c[/tex].
It means all the steps of factoring a quadratic expressions [tex]ax^2+bx+c[/tex] and [tex]x^2+bx+c[/tex] are same except the first step, i.e., divide the expressions [tex]ax^2+bx+c[/tex] by a.
2.
The given quadratic expression is
[tex]P(x)=2x^2+6x-20[/tex]
[tex](2x-4)(x+5)=2x(x+5)-4(x+5)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)[/tex]
[tex](x-2)(2x+10)=x(2x+10)-2(2x+10)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)[/tex]
The product of factors is equal to the given expression. It means both of these factorization are correct.
They are not complete because they can be factored further.
[tex](2x-4)(x+5)=2(x-2)(x+5)[/tex]
[tex](x-2)(2x+10)=(x-2)2(x+5)[/tex]
3.
If the factored form of a quadratic expression is defined as
[tex](x-a)(x-b)[/tex]
Then the related quadratic equation is
[tex](x-a)(x-b)=0[/tex]
[tex]x=a,b[/tex]
The roots of the quadratic equation are a and b.
The related function is
[tex]f(x)=(x-a)(x-b)[/tex]
The x-intercepts of the function are a and b because at x=a and x=b the value of function is 0.
The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.
It should be noted that in order to factorize the quadratic expression, one will have to divide it by a.
FactorizationThe factorization of the quadratic expression ax² + bx + c is different from factoring x² + bx + c as one has to first divide it by a.
Secondly, the factorization by the students isn't complete because they can be factored further.
Lastly, the relationship between the factors of a quadratic expression, the roots of the related quadratic equation is that the roots are the x-intercept of the related function.
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A drill instructor recorded the time in which each of 11 recruits completed an obstacle course both before and after basic training. to test whether any improvement occurred, the instructor would use a t distribution with 11 degrees of freedom.
In this problem, it is TRUE that the instructor would need to use a t distribution with 10 degrees of freedom to be able to test whether any improved occurred.
In order to check for its correctness, we need to measure the absolute difference between the mean value in two groups in a clinical trial by using the mean difference. Next, we check for the degrees of freedom. The degrees of freedom are the number of independent pieces of information that went into calculating the estimate.
You have to subtract 1 from
the number of items to be able to get the df for the estimate. Let’s check this
scenario:
Let’s say you need to find the mean weight loss for a low carbohydrate diet.
You can use 4 people and giving 3 degrees of freedom (4 -1 = 3), in turn, you
can also use 100 people with a df of = 99.
The value of a car decreases by 20% per year Mr. Singh for purchase is a $22,000 automobile what is the value of the car the end of the second year
Answer: the answer is $26,400
Step-by-step explanation:
six times the difference between a number and 4 equals the product of the number and -2. find the number.
I just want maybe a reteach for this.
*Solving problems with unknown numbers*