Answers
Answer:
(- 1, 4)
Step-by-step explanation:
The line x = 1 is a vertical line.
The point P(3, 4) is 2 units to the right of x = 1 ( 3 - 1 = 2 )
Hence the reflection will be 2 units to the left of x = 1, that is (1 - 2)
P'(- 1, 4)
The reflection of point P(3, 4) about the vertical line x = 1 is P'(-2, 4).
Reflection is a transformation that flips a figure over a line or a plane. It is a type of symmetry transformation that maintains the same shape and size of the object but changes its orientation with respect to the line or plane of reflection.
Given: Point P(3, 4).
The line x = 1 is a vertical line.
Now, let's find the reflection of point P(3, 4) about the vertical line x = 1:
To reflect a point about a vertical line, we reverse the sign of the x-coordinate while keeping the y-coordinate the same.
So, the reflection of P(3, 4) will be (-2, 4).
Hence, the reflection of point P(3, 4) is P'(-2, 4).
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Related Questions
The probability that an event will occur is 7/8 which of these best describes the likehood of the even occurring
Answers
Very Likely
think of it this way
you have a 7/8 chance of getting electrocuted by sticking your hand in the toaster
you have a 1/8 chance of this event not occurring when you stick said hand in the toaster
so its VERY LIKELY that you will be electrocuted if you stick your hand in a toaster
hope that helps!
The vertex of this parabola is at (2, -4). When the x-value is 3, the yvalue is -1. What is the coefficient of the squared expression in the parabola's equation?
Answers
Answer:
3
Step-by-step explanation:
Fill in the known values in the vertex form equation and solve for the coefficient.
y = a(x -h)^2 +k
-1 = a(3 -2)^2 -4 = a -4 . . . . fill in the values and simplify
3 = a . . . . . . . . . . . . . . . . . . .add 4
The coefficient of the squared expression is 3.
Final answer:
The coefficient of the squared term in the parabola's equation, given the vertex (2, -4) and a point (3, -1) on the parabola, is found to be 3 by substituting these values into the vertex form of a parabola's equation.
Explanation:
The student is asking how to determine the coefficient of the squared term in a parabola's equation, given the vertex and a point on the parabola. The standard form of a parabola's equation with vertex (h, k) is [tex]y = a(x - h)^2 + k,[/tex] where a is the coefficient in question. Knowing the vertex at (2, -4) and a point (3, -1) on the parabola, we can substitute these into the equation to find a.
Substituting the vertex into the equation gives us the form [tex]y = a(x - 2)^2 - 4.[/tex] Then we substitute the point (3, -1):
[tex]-1 = a(3 - 2)^2 - 4[/tex]
[tex]-1 = a(1)^2 - 4[/tex]
-1 + 4 = a · 1
a = 3
Therefore, the coefficient of the squared expression in the parabola's equation is 3.
There is a flu outbreak at your school that starts with 10 people. The number of ill students increases by 20% each hour. Write an exponential function to represent the total number of ill students, f(x), where x is the number of hours after the outbreak. How long does it take for at least 100 people to be ill with the flu?
a. About 10 hours
b. About 13 hours
c. About 20 hours
d. Not enough information
Answers
Answer:
d
Step-by-step explanation:
what does ' f ' represent?
The exponential function for the total number of ill students is [tex]f(x) = 10 * (1.20)^x,[/tex] where x is the number of hours after the outbreak. To reach at least 100 ill students, it takes about 13 hours. So correct answer is option B.
To represent the total number of ill students f(x) as an exponential function where x is the number of hours after the outbreak, we use the initial value of 10 people sick and an hourly increase rate of 20%. The function is: [tex]f(x) = 10 * (1 + 0.20)^x[/tex].
To find how long it takes for at least 100 people to be ill, we set f(x) \\>= 100 and solve for x:
[tex]10 * (1.20)^x \ > = 100\\(1.20)^x \ > = 10x\\\\log(1.20) \ > = \log(10)\\x > = \log(10) \\ \\log(1.20)\\x = 12.2[/tex]
Therefore, it takes about 13 hours for at least 100 people to be ill. So the answer is b. About 13 hours.
HEEEEELP ME ITS MATH I NEED THIS FAST PICTURE BELOW
Answers
See the attached picture for the answer.
Find the average value of y equals the square root of the quantity 64 minus x squared, on its domain.
Answers
Answer:
2π ≈ 6.283
Step-by-step explanation:
The average value of the function is the area under it, divided by the base. This function describes a semicircle of radius 8, so its area is ...
A = 1/2πr² = 1/2π·8² = 32π
The width of the base is the diameter of the semicircle, so is 16. Then the average value is ...
32π/16 = 2π . . . . . average value of y
Is the histogram symmetric, skewed right, or skewed left? Explain your answer .
Answers
Answer:
It’s skewed left
Step-by-step explanation:
The main mass of the graph is on the left. Follow the shape of the graph to find this
To determine if a histogram is symmetric or skewed, we assess the relative positions of the mean, median, and mode. A symmetric distribution has these three values approximately equal and the histogram's halves mirror each other. A right-skewed histogram has a tail extending right, whereas a left-skewed histogram's tail extends left.
Explanation:When analyzing whether a histogram is symmetric, skewed right, or skewed left, certain aspects of the distribution of data are considered. Symmetry in a histogram indicates that if a vertical line were drawn at some midpoint, the left and right sides would be mirror images. In a symmetric distribution, the mean, median, and mode are usually equivalent or very close to each other. To determine skewness, we look at the arrangement of these three measures: for a skew to the right, the mode is often less than the median, which is less than the mean. Conversely, a distribution skewed to the left typically has the mean less than the median, which is less than the mode.
A histogram for a right-skewed distribution will often show a tail extending to the right, indicating that larger numbers in the dataset are more spread out. On the other hand, a histogram with a left-skewed distribution will show a tail that extends to the left, which implies that the lower numbers are more spread out.
Which graph correctly solves the system of equations below? y = − x2 + 1 y = x2 − 4
A) quadratic graph opening up and quadratic graph opening down. They do not intersect.
B) quadratic graph opening up and quadratic graph facing down. They intersect at negative 2, negative 3 and 2, negative 3.
C) Two intercepting parabolas are shown, one facing downward and one facing upward. The downward facing parabola has a maximum at (0,1) and intercepts the x axis at 1 and negative 1. The upward facing parabola has a minimum at (0,-4) and intercepts the x axis at 2 and negative 2
D) two quadratic graphs opening up. They intersect at 0, negative 4.
Answers
Answer:
C) Two intercepting parabolas are shown, one facing downward and one facing upward. The downward facing parabola has a maximum at (0,1) and intercepts the x axis at 1 and negative 1. The upward facing parabola has a minimum at (0,-4) and intercepts the x axis at 2 and negative 2
Step-by-step explanation:
As shown in the attached, the graph of y = -x² +1 has its vertex at (0, 1) and opens downward. It crosses the x-axis at ±1. The graph of y = x² -4 has its vertex at (0, -4), opens upward, and crosses the x-axis at ±2. The description of this graph matches choice C.
Identify the graph that has a vertex of (1,-1) and a leading coefficient of a=2.
Answers
ANSWER
[tex]f(x) = 2 ({x - 1)}^{2} - 1[/tex]
EXPLANATION
The vertex form of a parabola has equation:
[tex]f(x) = a ({x - h)}^{2} + k[/tex]
where V(h,k) is the vertex of the parabola and 'a' is the leading coefficient.
From the question, we have that, the vertex is
[tex](1,-1)[/tex]
and the leading coefficient is
[tex]a= 2[/tex]
We substitute the vertex and the leading coefficient into the vertex form to get:
[tex]f(x) = 2 ({x - 1)}^{2} + - 1[/tex]
We simplify to get:
[tex]f(x) = 2 ({x - 1)}^{2} - 1[/tex]
The graph of this function is shown in the attachment.
The graph that has a vertex of (1,-1) and a leading coefficient of a=2 is a parabola.
Explanation:The graph that has a vertex of (1,-1) and a leading coefficient of a=2 is a parabola. The leading coefficient, which is the coefficient of the squared term, determines the nature of the parabola.
Since the leading coefficient is positive, the parabola opens upward. The equation of the parabola can be written in the form y = ax^2 + bx + c, where a represents the leading coefficient.
Therefore, the equation of the graph is y = 2x^2 - 4x + 1.
mary is solving the equation 5^x + 4 =11. her first steps are shown.
step 1. 5^x + 4 =11
step 2. 5^x =7
step3. In5^x=In7
which shows step 4?
A. In 5 = x In 7
B. x In 5 = In 7
C. ^-In 5 = In 7 * x
D. x = In 7 - In 5
Answers
Answer:
Option B. x In 5 = In 7
Step-by-step explanation:
we have
[tex]5^{x}+4=11[/tex]
step 1
[tex]5^{x}+4=11[/tex]
step 2
Subtract 4 both sides
[tex]5^{x}+4-4=11-4[/tex]
[tex]5^{x}=7[/tex]
step 3
Apply ln both sides
[tex]ln(5^{x})=ln(7)[/tex]
step 4
[tex]xln(5)=ln(7)[/tex]
step 5
Divide by ln(5) both sides
[tex]x=ln(7)/ln(5)[/tex]
Need help with a math question
Answers
Answer:
3%
Step-by-step explanation:
We are given the data of number of cars observed waiting in line at the beginning of 2 minute intervals between 3 and 5 p.m. on Friday.
We are to find the probability (in percent) that there is no one in line.
Sum of frequencies = 2 + 9 + 16 + 12 + 8 + 6 + 4 + 2 + 1 = 60
Frequency of no car in line = 2
P (no car in line) = 2 / 60 × 100 = 3.3% ≈ 3%
Find the area of the trapezoid.
Answers
For this case we have that by definition, the area of the trapezoid is given by:
[tex]A = \frac {1} {2} (B + b) * h[/tex]
Where:
B: It is the major base
b: It is the minor base
h: It's the height
Substituting the values according to the data of the figure:
[tex]A = \frac {1} {2} (2.1 + 0.9) * 1.3\\A = \frac {1} {2} (3) * 1.3\\A = \frac {1} {2} * 3.9\\A = 1.95[/tex]
Thus, the area of the trapezoid is[tex]1.95 m ^ 2[/tex]
ANswer:
Option B
PLEASE HELP ME WITH THIS MATH QUESTION
Answers
Answer:
r=4ft
h=8ft
Area of cyclender=?
by using formula,
A=πr²h
=22/7×4²×8
=402.28ft²Ans.
ANSWER
301.6 ft²
EXPLANATION
The surface area of a cylinder is calculated using the formula;
[tex]S.A = 2\pi \: r(r + h)[/tex]
From the diagram the height of the cylinder is 8 feet and the radius is 4 feet.
We substitute the values into the formula to obtain,
[tex]S.A = 2\pi \: 4(4+ 8)[/tex]
This simplifies to:
[tex]S.A = 8\pi \: (12)[/tex]
[tex]S.A = 96\pi[/tex]
Or
[tex]S.A = 301.6 {ft}^{2} [/tex]
to the nearest tenth.
Parametric Equations? How do you do them? I don't even know how to graph them and its so confusing because all the equations I put in say they're not written correctly???
Answers
Answer:
A rectangular equation, or an equation in rectangular form is an equation composed of variables like x and y which can be graphed on a regular Cartesian plane.
Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as "parameters." For example, while the equation of a circle in Cartesian coordinates can be given by r^2=x^2+y^2, one set of parametric equations for the circle are given by
x=r cost
y=r sin
Note that parametric representations are generally nonunique, so the same quantities may be expressed by a number of different parameterizations. A single parameter is usually represented with the parameter t, while the symbols u and v are commonly used for parametric equations in two parameters.Parametric equations provide a convenient way to represent curves and surfaces, as implemented, for example, in the Wolfram Language commands ParametricPlot[{x, y}, {t, t1, t2}] and ParametricPlot3D[{x, y, z}, {u, u1, u2}, {v, v1, v2}]. Unsurprisingly, curves and surfaces obtained by way of parametric equation representations are known as parametric curves and parametric surfaces, respectively.
Also if you dont know how to graph them you can Use DESMOS graphing calculator
Use long division or synthetic division to find the quotient of
Answers
Answer:
2x^2-x+1
Step-by-step explanation:
Answer:
2x² - x + 1
Step-by-step explanation:
[tex]\frac{(2x^{2}-x+1)(x+1)}{(x+1)} = 2x^{2}-x+1[/tex]
Why does the PCI require banks to protect customers’ card data?
A.
to protect banks from hackers and malware
B.
to help improve the cyber community
C.
to establish good practices in the banking community
D.
to protect consumers from online fraud and theft
Answers
Answer:
D. to protect consumers from online fraud and theft
Step-by-step explanation:
The point of protection of personal identifying and financial data is to prevent fraud and theft.
___
The reason why hackers and malware attack banks is to get to data that would enable fraud and theft. "Good practices" prevent such data compromise, so protecting customers from fraud and theft.
Answer:
Why does the PCI require banks to protect customers’ card data?
A.
to protect banks from hackers and malware
B.
to help improve the cyber community
C.
to establish good practices in the banking community
D.
to protect consumers from online fraud and theft
Step-by-step explanation:
#platofam
Determine the asymptotes of the function: y=x^3-5x^2+4x-25/x^2-4x+3
(horizontal, vertical or slant)
Answers
Answer:
Vertical A @ x=3 and x=1
Horizontal A nowhere since degree on top is higher than degree on bottom
Slant A @ y=x-1
Step-by-step explanation:
I'm going to look for vertical first:
I'm going to factor the bottom first: (x-3)(x-1)
So we have possible vertical asymptotes at x=3 and at x=1
To check I'm going to see if (x-3) is a factor of the top by plugging in 3 and seeing if I receive 0 (If I receive 0 then x=3 gives me a hole)
3^3-5(3)^2+4(3)-25=-31 so it isn't a factor of the top so you have a vertical asymptote at x=3
Let's check x=1
1^3-5(1)^2+4(1)-25=-25 so we have a vertical asymptote at x=1 also
There is no horizontal asymptote because degree of top is bigger than degree of bottom
There is a slant asympote because the degree of top is one more than degree of bottom (We can find this by doing long division)
x -1
--------------------------------------------------
x^2-4x+3 | x^3-5x^2+4x-25
- ( x^3-4x^2+3x)
--------------------------------
-x^2 +x -25
- (-x^2+4x-3)
---------------------
-3x-22
So the slant asymptote is to x-1
Answer: D
Step-by-step explanation:
EDGE 2021
PLEASE HELP! I'm on a time limit!! Identify the translation of the figure with the vertices L(1,−1), M(4,−3), and N(3,−5), along the vector ⟨2,5⟩.
L ′(3, 4), M ′(2, 6), N ′(5, 0)
L ′(3, 4), M ′(6, 2), N ′(5, 0)
L ′(1, 3), M ′(6, 2), N ′(0, 5)
N ′(3, 4), M ′(6, 2), N ′(5, −1)
Answers
Answer:
It's the second option.
Step-by-step explanation:
You add 2 to the x coordinate and 5 to the y coordinate.
So L' = (1 + 2, -1+5)
= (3, 4).
Answer:
L ′(3, 4), M ′(6, 2), N ′(5, 0)
two tables, congruent trapezoids, are placed together to make a corner desk, as shown
A. 8 square ft
B. 10 square ft
C. 16 square ft
D. 20 square ft
Answers
Answer:
D
Step-by-step explanation:
3 and 7 are the main factors so you add them and get 10 but since it’s two equilateral trapezoids then you get another 10 being 20 square feet.
Answer:
D) 20 square feet
Step-by-step explanation:
We are given two congruent isosceles trapezoids and placed together formed to make a corner of the desk.
We need to find the area.
We know that the area of a trapezoid = [tex]\frac{h}{2} [base 1+ base 2][/tex]
Where "h" is the height of the trapezoid.
Given: h = 2 ft, base 1 = 7ft and base 2 = 3ft
Now plug in these values in the above formula, we get
Area of a 1 trapezoid = [tex]\frac{2}{2} [7 + 3][/tex]
= 10 square feet
The two trapezoids are congruent.
So the area of the given figure = 2(10) = 20 square feet.
The graph of F(X), shown below, has the same shape as the graph of
G(x) = x2, but it is shifted up 1 unit. What is its equation?
Answers
Answer:
Option B. [tex]F(x)=x^{2}+1[/tex]
Step-by-step explanation:
we know that
[tex]G(x)=x^{2}[/tex]
This is the equation of a vertical parabola open upward wit vertex at (0,0)
The rule of the translation of G(x) to F(x) is equal to
(x,y) ----> (x,y+1)
therefore
The vertex of the function f(x) is the point (0,1) and the equation is equal to
[tex]F(x)=x^{2}+1[/tex]
The product is k2 – k + .
Answers
Come on, now. Incomplete question.
Answer:
is it late now
Step-by-step explanation:
What is the measure of RCD in the figure below?
Answers
Answer:
35 degrees.
Step-by-step explanation:
Triangles PDC and RDC are congruent ( By HL - The hypotenuse and one leg are equal).
Therefore m < RCD = 35 degrees.
The measure of of RCD in the figure is 35 degrees.
What is congruence?
Two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
In PDC and RDC,
angle DPC= angle DRC
CD=CD
PD=DR
By SAS congruence criteria,
ΔPDC ≈ ΔRDC
PCD= RCR= 35. (CPCT)
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HELP ASAP!! Lara starts from the school, which is 5 miles west and 7 miles north of the house. She travels 20 miles south, and then 15 miles east. What is her final position? What single translation vector moves her from her starting position to her final position?
Answers
Answer:
(10, −13); (15, −20)
Step-by-step explanation:
A hotel manager found that his gross recipients for the day, including a 7% sales tax, totaled to $3479.64. Find the amount of sales tax collected.
(I ask for someone to please, quickly, help me answer he question, I can't seem to properly do it myself.)
Answers
Answer:
$227.64
Step-by-step explanation:
The relevant relations are ...
sales + tax = total
tax = 7% × sales
Using the second equation, we can write sales in terms of the tax as ...
sales = tax/0.07
Substituting this into the first equation gives ...
tax/.07 + tax = total . . . . . substitute for sales
tax(1/0.07 + 1) = total . . . . factor out tax
tax ((1 +.07)/.07) = total . . . simplify to a single fraction
Multiply by the inverse of this fraction:
tax = .07/1.07 × total = (7/107)($3479.64)
tax = $227.64
The value of a collector’s item is expected to increase exponentially each year. the item is purchased for $500. after 2 years, the item is worth $551.25. which equation represents y, the value of the item after x years? y = 500(0.05)x y = 500(1.05)x y = 500(0.1025)x y = 500(1.1025)x
Answers
Answer:
[tex]y=500(1.05)^x[/tex]
Step-by-step explanation:
The standard form for an exponential equation is
[tex]y=a(b)^x[/tex]
We have 2 unknowns, a and b, but that's all good because we have 2 (x, y) coordinates we can utilize in order to find a and b. In our coordinate pair, x is the number of years gone by and y is the value after that number of years. The problem tells us that an item was purchased for $500. That translates to "before any time has gone by, the initial value of the item is $500". In other words, with x being time, no time has gone by, so x = 0. When x = 0, y = 500. (0, 500). Do the same for the next set of numbers. When x = 2 years gone by, the value is $551.25, so the coordinate is (2, 551.25). Now we use them to find a. Use the first coordinate:
[tex]500=a(b)^0[/tex]
Anything raised to the 0 power = 1, therefore:
[tex]500 = a(1)[/tex] and a = 500.
Now onto the next coordinate point using the a value we just found:
[tex]551.25 = 500(b)^2[/tex]
Divide both sides by 500 to get
[tex]1.1025=b^2[/tex]
so b = 1.05.
Now we have the values for a and b, so we fill in:
[tex]y = 500(1.05)^x[/tex]
PLEASE HELP!11 25 POINTS The volume of a right rectangular prism can be determined by multiplying the base area of the figure by the height. The volume of a right rectangular prism with a base area of 8 square inches is more than 64 cubic inches. The inequality 8h > 64 can be used to model the situation, where h represents the height of the figure. Which is a possible value of h?
a. 2
b.4
c.8
d.12
Answers
Answer:
12
Step-by-step explanation:
The only possible answer if 12 because all of the other choices come to the conclusion that 8h ≤ 64
if h=12 then 8h= 8 * 12 = 96 > 64
The value of h is 12.
What is the volume of a rectangular prism?Multiply the length, width, and height of a rectangular prism to determine its volume. Cubic measurements are used to express volume.
Given,
The only possible answer is 12 because all of the other choices come to the conclusion that 8h ≤ 64
if h=12 then 8h= [tex]8 * 12[/tex] = 96 > 64
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Graph the following system of linear inequalities. Identify at least two points in the solution: y < 5 - 2x | x + 5y > -7
Answers
Answer:
(1,2) and (2,-1)
Step-by-step explanation:
we have
[tex]y< 5-2x[/tex] ----> inequality A
The solution of the inequality A is the shaded area below the dashed line [tex]y=5-2x[/tex]
[tex]x+5y>-7[/tex] ---->inequality B
The solution of the inequality B is the shaded area above the dashed line [tex]x+5y=-7[/tex]
The solution of the system of inequalities is the triangular shaded area between the two dashed lines
If a ordered pair is a solution of the system of inequalities, then the ordered pair must lie on the shaded area
Two points in the solution are(1,2) and (2,-1)see the attached figure
Find the distance between these points. R(-1, 0), S(8, 6) √(26) √(85) 3√(13)
Answers
Answer:
[tex]|RS|=3\sqrt{13}[/tex] units.
Step-by-step explanation:
The distance between the two given points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] is given by:
[tex]d=\sqrt{(x_2-x-1)^2+(y_2-y_1)^2}[/tex]
We want to find the distance between R(-1, 0) and S(8, 6).
We plug in these points into the distance formula to get:
[tex]|RS|=\sqrt{(8--1)^2+(6-0)^2}[/tex]
[tex]|RS|=\sqrt{(9)^2+(6)^2}[/tex]
[tex]|RS|=\sqrt{81+36}[/tex]
[tex]|RS|=\sqrt{117}[/tex]
[tex]|RS|=3\sqrt{13}[/tex] units.
If the volume of the rectangular prism is represented by 6x2 – 2x + 8 and the base area is 2x – 4, which expression represents the height?
Answers
The expression that represents the height of the rectangular prism is 3x + (10 / (2x - 4)) found by dividing the given volume expression by the base area expression.
To find the expression that represents the height of the rectangular prism, we need to rearrange the formula for the volume of a prism, which is Volume = Base Area x Height. Given the volume of the rectangular prism as 6x^2
- 2x + 8 and the base area as 2x - 4, we divide the volume by the base area to find the height:
Height = Volume / Base Area
Height = (6x^2 - 2x + 8) / (2x - 4)
This simplifies to:
Height = 3x + (10 / (2x - 4))
Therefore, the expression that represents the height is 3x + (10 / (2x - 4)).
An aeroplane at an altitude of 200m observes the angle of depression of opposite points on the two banks of a river to be 45 and 60 find the widht of river
Answers
Answer:
84.5 m
Step-by-step explanation:
It is often helpful to draw a diagram for word problems involving geometric relationships. One for this problem is shown below.
The mnemonic SOH CAH TOA reminds you of the relationship between sides of a right triangle:
Tan = Opposite/Adjacent
Here we're given angles of depression measured from the horizontal (as shown in the diagram), but it is more convenient to use angles measured from the vertical. In particular, ∠BAO is the complement of 60°, and its tangent is the ratio OB/OA:
tan(30°) = OB/OA
OB = (200 m)·tan(30°) ≈ 115.47 m . . . . . . multiply by OA, use OA=200 m
Likewise, we have ...
OC = (200 m)·tan(45°) = 200 m
Then the width of the river is the difference between these values:
BC = OC -OB = 200 m - 115.47 m = 84.53 m
Without using technology, describe the end behavior of f(x) = −3x4 + 7x2 − 12x + 13.
Answers
Following are the description on the function behavior:
Given:
[tex]\bold{f(x) = -3x^4 + 7x^2 - 12x + 13}[/tex]
To find:
Function behavior=?
Solution:
We use Power and Polynomial Functions features in the absence of technology. As the function [tex]\bold{f(x) = -3x^4 + 7x^2 -12x + 13}[/tex]
For final behaviour of power functions of such form[tex]\bold{f(x)=ax^n}[/tex] wherein n is a non-negative integer depends on the power and the constant.
So, the leading term, [tex]\bold{f(x)=-3x^4}[/tex]
When the negative constant and even power are:
[tex]\to x \to \infty\\\\\to f(x) \to -\infty[/tex]
At
[tex]x \to -\infty\\\\f(x) \to -\infty[/tex]
Therefore, the final answer is "Down on the left down on the right "
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The end behavior of [tex]\( f(x) = -3x^4 + 7x^2 - 12x + 13 \)[/tex] is described as "Down on the left, down on the right," The correct answer is option a) Down on the left, down on the right.
To determine the end behavior of the polynomial [tex]\( f(x) = -3x^4 + 7x^2 - 12x + 13 \)[/tex] without using technology, we analyze the leading term, which dominates the behavior of the function as x approaches positive or negative infinity.
1. Identify the leading term: The leading term of [tex]\( f(x) \) is \( -3x^4 \)[/tex].
2. Consider the degree and leading coefficient:
- The degree of the polynomial is 4.
- The leading coefficient (coefficient of the term with the highest power of [tex]\( x \)) is \( -3 \)[/tex].
3. Determine the end behavior:
- As [tex]\( x \to +\infty \), \( -3x^4 \)[/tex] approaches [tex]\( -\infty \)[/tex] because [tex]\( x^4 \)[/tex] grows much faster than the negative coefficient affects it. Therefore, [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( -3x^4 \)[/tex] also approaches [tex]\( -\infty \)[/tex] for the same reason. Hence, [tex]\( f(x) \to -\infty \)[/tex].
4. Conclusion: Based on the analysis:
- The polynomial [tex]\( f(x) = -3x^4 + 7x^2 - 12x + 13 \)[/tex] decreases to [tex]\( -\infty \)[/tex] as x goes to both positive and negative infinity.
Therefore, the end behavior of [tex]\( f(x) \)[/tex] is described as "Down on the left, down on the right", which corresponds to option a). This indicates that the graph of [tex]\( f(x) \)[/tex] starts high on the left and continues downward indefinitely in both directions.
Complete question : Without using technology, describe the end behavior of f(x) = −3x4 + 7x2 − 12x + 13.
a Down on the left, down on the right
b Down on the left, up on the right
c Up on the left, down on the right
d Up on the left, up on the right