Without plotting any points other than intercepts, draw a possible graph of the following polynomial:
f(x) = (x + 8)3(x + 6)2(x + 2)(x − 1)3(x − 3)4(x − 6).
Answers
0 = (x + 8)3(x + 6)2(x + 2)(x − 1)3(x − 3)4(x − 6)
There are 6 intercepts:
x+8=0 ---> x= -8
x+6=0 ---> x=-6
x+2=0 ---> x=-2
x-1=0 ---> x=1
x-3=0 ---> x=3
x-6=0 ---> x=6
The 6 intercepts are plotted as shown in the picture. The tool I used was Microsoft Excel. This is one possible sketch of the graph. The trend could move in any ways as long as it passes these 6 intercepts.
Final answer:
To sketch the polynomial graph without plotting points beyond intercepts, identify the x-intercepts from the factors, note the behavior at these points due to the powers, and connect them considering the end behavior akin to y = x^13.
Explanation:
To sketch a possible graph of the given polynomial f(x) = (x + 8)^3(x + 6)^2(x + 2)(x - 1)^3(x - 3)^4(x - 6), we can use its intercepts and the behavior near these intercepts. The intercepts are the x-values for which f(x) = 0. These are at x = -8, x = -6, x = -2, x = 1, x = 3, and x = 6.
Since these factors are raised to different powers, the graph will behave differently at each intercept. Also, since the highest power in the polynomial is 13 (3+2+1+3+4+1), the end behavior of f(x) as x approaches positive or negative infinity will resemble that of y = x^13.
Here's a step-by-step guide for sketching the graph:
Plot the x-intercepts on the x-axis: (-8, 0), (-6, 0), (-2, 0), (1, 0), (3, 0), (6, 0).
For each intercept, note that if the factor has an odd power, the graph crosses the x-axis, and if it has an even power, the graph touches the x-axis and turns back.
At x = -8, since the power is 3 (odd), the graph crosses the axis and behaves similarly to y = x^3, starting from the bottom.
At x = -6 and x = -2, the powers are 2 (even), so the graph will just touch the axis and turn back.
At x = 1, the power is 3 (odd), so the graph will cross the axis with the behavior of y = x^3.
At x = 3, the power is 4 (even), hence the graph touches the axis and turns back.
At x = 6, as the factor is raised to the first power (odd), the graph crosses the x-axis.
Finally, connect these points with smooth curves, keeping in mind the end behavior of the graph will rise on both ends because the degree is 13, an odd number.
Remember that the final graph should show the pattern of rising and falling appropriately between these intercepts based on the powers of each factor.
Related Questions
The function f(x) = –x2 + 28x – 192 models the hourly profit, in dollars, a shop makes for selling sodas, where x is the number of sodas sold and f(x) is the amount of profit. Part A: Determine the vertex. What does this calculation mean in the context of the problem? (5 points) Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)
Answers
[tex]\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{llccll} y = &{{ -1}}x^2&{{ +28}}x&{{ -192}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)[/tex]
check the picture below.
b)
[tex]\bf f(x)=-x^2+28x-192\implies 0=-x^2+28x-192 \\\\\\ x^2-28x+192=0\implies (x-16)(x-12)=0\implies x= \begin{cases} 16\\ 12 \end{cases}[/tex]
what does this mean? check the picture below, notice the x-intercepts points.
Factor the expression
4b^2+28b+49
Answers
Which term best describes a proof in which you assume the opposite of what you want to prove?
Answers
Answer:
Contradictory Statement
Step-by-step explanation:
In contradictory statement , we basically describes a proof by assuming opposite of what we want to prove .
For example :
Prove that [tex]\sqrt{2}[/tex] is irrational .
Solution :
We will prove this by contradiction .
Let if possible [tex]\sqrt{2}[/tex] is rational .
[tex]\sqrt{2}=\frac{p}{q}[/tex] where p and q are integers and coprime such that [tex]q\neq 0[/tex]
On squaring both sides , we get
[tex]2q^2=p^2\\\Rightarrow 2|p^2\\\Rightarrow 2|p[/tex]
we get ,
p=2r
On squaring both sides, we get
[tex]p^2=4r^2\\\Rightarrow 2q^2=4r^2\\\Rightarrow q^2=2r^2\\\Rightarrow 2|q^2\\\Rightarrow 2|q[/tex]
So, 2 divides p and q which is a contradiction to the fact that p and are coprime .
Therefore, [tex]\sqrt{2}[/tex] is irrational .
a tower casts a 450 ft shadow at the same time that a 4 ft child casts a 6 ft shadow. Write and solve a proportion to find the height of a tower
Answers
Answer:
[tex]x=300[/tex]
Step-by-step explanation:
Set up the proportion;
[tex]\frac{x}{450} =\frac{4}{6}[/tex]
then cross multiply;
[tex]6x=450[/tex] · [tex]4[/tex]
[tex]6x=1800[/tex]
[tex]x=\frac{1800}{6} =300[/tex]
[tex]x=300[/tex]
rationalize the denominator. write it in simplest terms
3
------
√12x
Answers
[tex]\bf \cfrac{3}{\sqrt{12x}}\cdot \cfrac{\sqrt{12x}}{\sqrt{12x}}\implies \cfrac{3\sqrt{12x}}{\sqrt{(12x)^2}}\implies \cfrac{3\sqrt{12x}}{12x}\implies \cfrac{\sqrt{12x}}{4x}[/tex]
An upscale resort has built its circular swimming pool around a central area that contains a restaurant. The central area is a right triangle with legs of 60 feet, 120 feet, and approximately 103.92 feet. The vertices of the triangle are points on the circle. The hypotenuse of the triangle is the diameter of the circle. The center of the circle is a point on the hypotenuse (longest side) of the triangle. The building permit the resort obtained requires that the resort state how much water the pool will hold so the city can manage the resort’s water rights effectively.
a.)What is the area of the largest section of the pool? Explain if you feel this area would be large enough to add a waterslide.
b.)How much water do you need to fill just the pool without the fish tank, if the average depth is 4 feet?
Answers
The largest section of the pool is 11307.2 square feet and it seems feasible to add a water slide based on this area. In order to fill the pool, considering that the average depth is 4 feet, approximately 338511.6 gallons of water would be needed.
Explanation:We will first find the area of the circular pool, keeping in mind that the hypotenuse of the triangle serves as the diameter. We will use the formula for the area of a circle, A = πr², where r is the radius of the circle. Here, the radius is half of the hypotenuse so it's 120 feet / 2 = 60 feet.
This makes the area of the circle A = π * 60² = 3600π square feet. This, minus the area of the triangle which can be calculated by (1/2) * base * height = (1/2) * 60 * 103.92 = 3117.6 square feet. So, the largest section of the pool is 3600π - 3117.6 = 11307.2 square feet.
Additional structures such as a water slide could be considered based on the remaining area. Due to size of the pool, it seems feasible to add a water slide without significantly disrupting the swimming space. However, additional calculations considering the base area of the slide would be required for a definitive answer.
The amount of water needed to fill the pool is calculated by multiplying the volume of the pool by the weight of the water. If the average depth of the pool is 4 feet, the volume of the pool (ignoring the central triangle) is 11307.2*4 = 45228.8 cubic feet. As 1 cubic foot of water equals roughly translates to 7.48 gallons, the pool would require approximately 45228.8 * 7.48 = 338511.6 gallons of water.
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determine which of the following logarithms is condensed correctly
Answers
alogb = log b^a
log a + log b = log (ab)
log a - log b = log (a/b)
for letter A,
xlogb r + logb s - logb t = logb [(rs)^x]/t
is wrong because the s part from logb s has no x before the logb
for letter B,
xlogb r + xlogb s - logb t = logb (rs/t)^x
is wrong because the t part from logb t has no x before logb
for letter C,
logbr - xlogb s + xlogb t = logb (r)/(st)^x
is correct because after the minus sign, condensing it would form a fraction, the plus sign will form a multiplication and both s and t are raised to the power of x.
So the answer is letter C.
A herd of 39 horses has 33 white horses some black horses. What is the ratio of black horses to white horses?
Answers
There is 6 black horses, so 6/39.
Solve the inequality.
Answers
[tex]\frac{2}{5} \geq x - \frac{4}{5}[/tex]
You get the x variable on it's own by undoing the operations done to it.
For example, if x is being multiplied by 5, you undo the multiplication operation by using the inverse of multiplication. Which is division.
We need to add [tex]\frac{4}{5}[/tex] to both sides of the inequality to undo the subtraction operation done to x.
[tex]\frac{2}{5} + \frac{4}{5} \geq x - \frac{4}{5} + \frac{4}{5} \\ \\ \frac{6}{5} \geq x[/tex]
Convert the improper fraction into a mixed number.
[tex]\frac{6}{5} = 1 \frac{1}{5}[/tex]
So, D 1 1/5 ≥ x is the answer.
When simplified, the expression (x ^1/8) (x^3/8) is 12. Which is a possible value of x?
Answers
x=144
Approximately 7% of people are left-handed. if two people are selected at random, what is the probability of p(one is right-handed and the other is left-handed)
Answers
The required probability that one is right-handed and the other is left-handed is 217/1650.
Given that,
Approximately 7% of people are left-handed. if two people are selected at random, what is the probability of p(one is right-handed and the other is left-handed) is to be determined.
What is probability?
Probability can be defined as the ratio of favorable outcomes to the total number of events.
Here,
Total number of left-handed people out of 100 = 7
Total number of right-handed people out of 100 = 100 - 7 = 93
Now,
Probability of picking 2 person(one is right-handed and the other is left-handed) = 7 / 100 (93/99) = 217/1650.
Thus, the required probability that one is right-handed and the other is left-handed is 217/1650.
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Draw a graph of the rose curve.
r = 2 sin 3θ, 0 ≤ θ ≤ 2π
Answers
help help help help help please
Answers
x - 2(x + 10) = 12 what's x
Answers
First, let's rewrite our problem.
x - 2(x + 10) = 12, solve for x.
To start us off, we need to apply the distributive property to the left side of the equation "-2(x + 10)", as since there is no sign between the number and the parenthesis, it is implied that we must multiply.
To distribute, we multiply the number outside of the parenthesis by all numbers inside the parenthesis.
For example;
2(1 + 3)
2(1) + 2(3)
2 + 6
8.
Now that we (hopefully) understand our concept, let's proceed to our equation to solve for x.
x - 2(x + 10) = 12
Apply the Distributive Property.
-2(x) - 2(!0)
-2x - 20.
We now have:
x - 2x - 20 = 12
Combine like-terms
-x - 20 = 12
Add 20 to both sides to isolate -x.
-20 + 20 = 0
12 + 20 = 32
Now we are left with:
-x = 32
However, we are not done as we still have x being multiplied by -1 (-x just means -1x). To get rid of the -1, we need to divide both sides by -1 to cancel them out.
-x / -1 = x
32 / -1 = -32
x = -32 is your solution.
I hope this helps!
If you add an expression to both sides of an equation, the resulting equation will have the same solution set as the original equation. In other words, they will be equivalent. This is true for all operations. As long both sides are treated the same, the equation will stay balanced.
You will also need to know how to combine like terms. But what are like terms to begin with? Like terms are defined as two terms having the same variable(s) (or lack thereof) and are raised to the same power. In mathematics, something raised to the first power stays the same. So, 5x and 10x are like terms because they both have the same variable and are raised to the first power. You don’t see the exponents because it doesn’t change the value of the terms.
To combine like terms, simplify add the coefficients and keep the common variable(s) and exponent.
The distributive property is another important rule you will need to understand. The distributive property is used mostly for simplifying parentheses in expressions/equations.
For example, how would you get rid of the parentheses here?
6(x + 1)
If there wasn’t an unknown in between the parentheses, you could just add then multiply. That is what the distributive property solves. The distributive property states that a(b + c) = ab + ac
So, now we can simplify our expression.
6(x + 1) = 6x + 6
To sum up how to solve algebraic equations:
Use the distributive property if needed
Combine like terms on both sides of the equation if needed
If you do one operation on one side of the equation, do the same on the opposite side
x - 2(x + 10) = 12
x - 2x - 20 = 12 <-- Using the distributive property
-x - 20 = 12 <-- Combing like terms
-x = 32 <-- Add 20 to both sides
x = -32 <-- Divide both sides by -1
So, x is equal to -32.
what is the product? 3x^5 (2x^2+4x+1)
Answers
Answer:
The product of the given polynomial [tex]3x^5(2 x^2+4x+1)[/tex] is [tex]6x^7+12x^6+3x^5[/tex]
Step-by-step explanation:
Given: Polynomial [tex]3x^5(2 x^2+4x+1)[/tex]
We have to find the product of the given polynomial [tex]3x^5(2 x^2+4x+1)[/tex]
Consider the given polynomial [tex]3x^5(2 x^2+4x+1)[/tex]
Apply distributive rule, [tex]a(b+c)=ab+ac[/tex]
Multiply [tex]3x^5[/tex] with each term in brackets, we have,
[tex]=3x^5\cdot \:2x^2+3x^5\cdot \:4x+3x^5\cdot \:1[/tex]
[tex]=3\cdot \:2x^5x^2+3\cdot \:4x^5x+3\cdot \:1\cdot \:x^5[/tex]
Apply exponent rule, [tex]a^b\cdot \:a^c=a^{b+c}[/tex]
Simplify, we have,
[tex]=6x^7+12x^6+3x^5[/tex]
Thus, The product of the given polynomial [tex]3x^5(2 x^2+4x+1)[/tex] is [tex]6x^7+12x^6+3x^5[/tex]
Simplify
[tex] \frac{ \frac{1}{x+h} + \frac{1}{x} }{x} [/tex]
Answers
How can you use a number line to compare and order numbers?
Answers
Take the value '0' for example, the value on the right of zero is '1', which is a value bigger than zero, and the value to the left of '0' is '-1', which is a value smaller than zero.
the "h" value moves the graph up or down true or false ?
Answers
The answer to this question is FALSE.
Answer:
False
Step-by-step explanation:
Suposse that we are given a function f(x) and a constant value h.
1. Case:
If we take the function g(x)=f(x)+h, then the graph of the function g(x) will be the graph of the funcion f(x) moved up or down.
2.Case:
If we take the function g(x)=hf(x), then the graph of the function g(x) will be the graph of the function f(x) just taller or shorter.
3.Case:
If we take the function g(x)=f(x-h), then the graph of the function g(x) will be the graph of the fuction f(x) moved horizontally.
4. Case:
If we take the function g(x)=f(hx), then the graph of the function g(x) will be tha graph of the function f(x) wither or thiner.
For example:
If we take f(x)=sin(x) and h=2. Then, if we take g(x)=sin(2x) then f(0)=g(0)=0, which means that the graph of the functiction is not moved up or down. However, f(π/2)=sin(π/2)=1 and g(π/2)=sin(π)=0 which gives us a hint that the graph of the function became thiner.
Hillary age is 3 years greater than 2/3 times Lilly's age if Hillary is 19 years old how old is Lilly
Answers
Hillary = 3+2/3X
19=3+2/3X
16=2/3X
X= 16 / 2/3 = 16/1* 3/2 = 48/2
X=24
Lilly is 24
Hillary age = 19 years old
Hillary age is 3 years greater than 2/3 times Lilly's age =
2/3x + 3 = 19
2/3x = 16
x = 16 * 3/2
x = 24
answer
Lily is 24 years old
Rectangle R has varying length l and width w but a constant perimeter of 4 ft. A. Express the area A as a function of l. What do you know about this function? B. For what values of l and w will the area of R be greatest? Give an algebraic argument. Give a geometric arguement.
Answers
l = length of the rectangle
w = width of the rectangle
P = 4 ft, constant perimeter
Because the given perimeter is constant,
2(w + l) = 4
w + l = 2
w = 2 - l (1)
Part A.
The area is
A = w*l
= (2 - l)*l
A = 2l - l²
This is a quadratic function or a parabola.
Part B.
Write the parabola in standard form.
A = -[l² - 2l]
= -[ (l -1)² - 1]
= -(l -1)² + 1
This is a parabola with vertex at (1, 1). Because the leading coefficient is negative the curve is downward, as shown below.
The maximum value occurs at the vertex, so the maximum value of A = 1.
From equation (1), obtain
w = 2 - l = 2 - 1 = 1.
The maximum value of the area occurs when w=1 and l=1 (a square).
Answer:
The area is maximum when l=1 and w=1.
The geometric argument is based on the vertex of the parabola denoting maximum area.
Find all solutions in the interval [0, 2π).
sin^2 x + sin x = 0
Answers
sin x( sin x + 1) = 0
sin x = 0 , or sinx + 1 = 0 giving sin x = -1
when sin x = 0 x = 0 , pi
when sin x = -1, x = pi + pi/2 = 3pi/2
solutions in given interval are 0,pi and 3pi/2
find the area of triangle QRS.
Answers
Answer: 140 square units.
Step-by-step explanation:
The area of triangle with vertices [tex](x_1,y_1),(x_2,y_2)\ and\ (x_3,y_3)[/tex] is given by
[tex]\text{Area}=\frac{1}{2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2][/tex]
Then area of triangle QRS with vertices (6,10), (2,-10) and (-9,5) is given by :-
[tex]\\\\\Rightarrow\text{Area}=|\frac{1}{2}[6(-10-5)+2(5-10)-9(10-(-10))]|\\\\\Rightarrow\text{Area}=|\frac{1}{2}[6(-15)+2(-5)+-9(20)]|\\\\\Rightarrow\text{Area}=|\frac{1}{2}[280]|\\\\\Rightarrow\text{Area}=140\text{ square units}[/tex]
Answer:
the answer is 140. I did the assignment
If you place a cup of 210 degree Fahrenheit coffee on a table in a room that is 68 degrees fareinheit, and ten minutes later, it is 200 degrees fareinheit. How long will it be before the coffee is 180 degrees fareinheit? Use newtons law of cooling
Answers
Tf=Ts+(Ti-Ts)e^(-kt) where Tf is temp at time t, Ts is temp of surroundings, Ti is temp of object/fluid. So we need to find k first.
200=68+(210-68)e^(-10k)
132=142e^(-10k)
132/142=e^(-10k)
ln(132/142)=-10k
k=-ln(132/142)/10
k≈0.0073 so
T(t)=68+142e^(-0.0073t) so how long until it reaches 180°?
180=68+142e^(-0.0073t)
112=142e^(-0.0073t)
112/142=e^(-0.0073t)
ln(112/142)=-0.0073t
t= -ln(112/142)/(0.0073)
t≈32.51 minutes
Final answer:
According to Newton's Law of Cooling, it will take approximately 10.76 minutes for the coffee to cool from 210 degrees Fahrenheit to 180 degrees Fahrenheit when placed in a room with a temperature of 68 degrees Fahrenheit.
Explanation:
According to Newton's Law of Cooling, the rate at which an object cools is proportional to the temperature difference between the object and its surroundings. We can use this law to determine how long it will take for the coffee to reach a temperature of 180 degrees Fahrenheit.
First, we need to find the temperature difference between the coffee and its surroundings. The initial temperature of the coffee is 210 degrees Fahrenheit and the room temperature is 68 degrees Fahrenheit. Therefore, the initial temperature difference is 210 - 68 = 142 degrees Fahrenheit.We also know that after 10 minutes, the coffee's temperature has decreased to 200 degrees Fahrenheit. So the new temperature difference is 200 - 68 = 132 degrees Fahrenheit.Now we can set up a proportion to find the time it takes to reach a temperature of 180 degrees Fahrenheit. The ratio between the initial temperature difference and the time it takes for the coffee to cool to 180 degrees Fahrenheit is equal to the ratio between the new temperature difference and the total time it takes for the coffee to cool to 180 degrees Fahrenheit. Let's call the total time T: (142 / T) = (132 / 10). Cross-multiplying, we get 1420 = 132T. Dividing both sides by 132, we find that T = 10.76 minutes.Therefore, it will take approximately 10.76 minutes for the coffee to reach a temperature of 180 degrees Fahrenheit.
Are there any rational numbers between 0.4 and 4/9
Answers
4/9 = 20/45
so the rational number in between can be 19/45
06.01 LC)
Four graphs are shown below:
Which graph represents a positive nonlinear association between x and y?
Graph A
Graph B
Graph C
Graph D
Answers
Graph D is the correct answer because it represents a POSITIVE and EXPONENTIAL (non-linear) relationship.
Answer:
d
Step-by-step explanation:
In a random sample of 75 individuals, 52 people said they prefer coffee to tea. 99.7% of the population mean is between 84/80/74 % and 64/58/54%. (Choose the option closest to your answer.)
Answers
He lake is 60 miles away, and rene is driving at 40 miles per hour the entire time, how long will it take her to get to the lake
Answers
divide distance by speed
60/40 = 1.5
it will take 1 1/2 hours ( 90 minutes)
Tim bought a soft drink for 2 dollars and 5 candy bars. He spent a total of 22 dollars. How much did the candy bar costs?
Answers
Subtract 2 from both sides
5x=20
Divide both sides by 5
x=4
Final answer: $4.00
PLEASE HELP!!! The volume in cubic feet of a box can be expressed as , or as the product of three linear factors with integer coefficients. The width of the box is x-2.
Factor the polynomial to find linear expressions for the height and the length. Show your work.
Answers
x(x^2-6x+8)
x(x^2-2x-4x+8)
x(x(x-2)-4(x-2))
x(x-2)(x-4)
Since we are told that the width is (x-2) the length and height are:
x and (x-4)
To find linear 12/10/2023 for the height and length of the box, factor the given polynomial (x - 2)(...)
Explanation:To find linear expressions for the height and length of the box, we need to factor the given polynomial. The volume of the box can be expressed as (x - 2)(?)(?), where ? represents the linear expressions for height and length. Since we have two unknowns, we need to find two factors that when multiplied together give us the given polynomial.
To factor x^2 - 2x, we need to find two numbers that multiply to give -2 and add up to give -2. The factors are -1 and 2, so we can write the polynomial as (x - 1)(x - 2). Therefore, the linear expressions for the height and length are (x - 1) and (x - 2) respectively.
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Which two consecutive whole numbers is the value of √21 between?
A.4 and 5
B.10 and 11
C.21 and 22
D.42 and 43
Answers
4 and 5
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Explanation:
______________________________
Looking at the answers:
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we know that 4² = 16 ; and 5² = 25 ;
So: √16 = 4 ; and √25 = 5.
So: √21 must be between √16 and √25 ;
That is: √21 must be between 4 and 5 ; which is answer choice: [A] .
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