Create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. (Round your answers to four decimal places. If an answer does not exist, enter DNE.)
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Answer:
Explanations:
Given the limit of the function expressed as:
[tex]\begin{gathered} \lim _{n\to0}\frac{\sin8x}{x} \\ f(x)=\frac{\sin 8x}{x} \end{gathered}[/tex]First, we need to create a table for the given values in the table:
If x = -0.1
[tex]\begin{gathered} f(-0.1)=\frac{\sin8(-0.1)}{-0.1} \\ f(-0.1)=\frac{\sin(-0.8)}{-0.1} \\ f(-0.1)=0.1396 \end{gathered}[/tex]If x = -0.01
[tex]\begin{gathered} f(-0.01)=\frac{\sin8(-0.01)}{-0.01} \\ f(-0.01)=\frac{\sin(-0.08)}{-0.01} \\ f(-0.01)=0.1396 \end{gathered}[/tex]If x = -0.001
[tex]\begin{gathered} f(-0.001)=\frac{\sin8(-0.001)}{-0.001} \\ f(-0.001)=\frac{\sin(-0.008)}{-0.008} \\ f(-0.001)=0.1396 \end{gathered}[/tex]From the values above, we can conclude that the values of f(x) will all tend to be 0.1396 for the positives values of x
Therefore, we can conclude that as you approach the value 0 from the positive and negative directions, they approach the same value, hence the limit does exist.
Related Questions
determine wether the point is a solution of the system. (-1,-2) 5x-2y=-1 x-3y=5
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Step 1
Given; The system of equation;
[tex]\begin{gathered} 5x-2y=-1--(a) \\ x-3y=5---(b) \\ \text{Required; To know if the point (-1,-2) is a solution to the system} \end{gathered}[/tex]Step 2
Input x=-1 and y=-2 in both equations and find if they will give -1 and 5 respectively.
[tex]\begin{gathered} 5(-1)-2(-2)=-5+4=-1 \\ -1-3(-2)=-1+6=5 \end{gathered}[/tex]Since both equations gave us -1 and 5 respectively when we input x=-1 and y=-2, then we can conclude that the point (-1,-2) is a solution to the system.
2. There were 132 students on the field trip. The students were divided into as many groups of 8 as possible. One group was smaller. How many students were in the smaller group? A 17 students B. 16 students C. 8 students D. 4 students
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So, there were 132 students divided in groups of 8.
If we divide:
So, there will be 8 groups of 16 and 1 group of 4 students. (The smaller group).
72 inches=___ yards please and thank you for your help
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We know that:
[tex]1yd=36in\text{.}[/tex]Then:
[tex]1in=\frac{1}{36}yd\text{.}[/tex]Therefore:
[tex]72in=72\cdot(\frac{1}{36}yd)\text{.}[/tex]Simplifying the above result we get:
[tex]72\cdot\frac{1}{36}yd=2yd\text{.}[/tex]Answer:
[tex]72\text{inches}=2\text{yards.}[/tex]Find the volume of a cylinder whose base has a radius of 3 inches and whose height is 12.5 inches. Use π = 3.14 and round your answer to the nearest tenth37.5 in^3333.8 in^3353.3 in^3421.8 in^3
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Answer:
353.3 in^3
Explanation:
Given:
The radius of the base of the cylinder (r) = 3 inches
The height of the cylinder (h) = 12.5 inches
pi = 3.14
To find:
The volume of the cylinder
We'll use the below formula to determine the volume(V) of the cylinder;
[tex]V=\pi r^2h[/tex]Let's go ahead and substitute the given values into the formula and solve for V;
[tex]V=3.14*3^2*12.5=3.14*9*12.5=353.3\text{ in}^3[/tex]So the volume of the cylinder is 353.3 in^3
A rectangular park is 60 meters wide and 105 meters long. Give the length and width of another rectangular park that has the same perimeter but a smaller area.
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First we find the area of the first park
[tex]\begin{gathered} A=w\times l \\ A=60\times105 \\ A=6300 \end{gathered}[/tex]area is 6300 square meters
now find the perimeter
[tex]\begin{gathered} P=2w+2l \\ P=2(60)+2(105) \\ P=330 \end{gathered}[/tex]perimeter is 330 meters
now we need to write equations to find the measures of the another park and we can write from the statements
has the same perimeter
[tex]2w+2l=330[/tex]but a smaller area
then we choose an area smaller than 6300, for example 6000
[tex]w\times l=6000[/tex]now we have two equations and two unknows
[tex]\begin{gathered} 2w+2l=330 \\ w\times l=6000 \end{gathered}[/tex]then we can solve a unknow from one equation and replace on the other
I will solve w from the first equation and replace on second
[tex]\begin{gathered} 2w=330-2l \\ w=\frac{330-2l}{2} \\ \\ w=165-l \end{gathered}[/tex][tex]\begin{gathered} w\times l=6000 \\ (165-l)\times l=6000 \\ 165l-l^2=6000 \end{gathered}[/tex]rewrite the equation
[tex]l^2-165l+6000=0[/tex]and use quadratic formula to solve L
[tex]\begin{gathered} l=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \\ l=\frac{-(-165)\pm\sqrt[]{(-165)^2-4(1)(6000)}}{2(1)} \\ \\ l=\frac{165\pm\sqrt[]{27225-24000}}{2} \\ \\ l=\frac{165\pm\sqrt[]{3225}}{2} \end{gathered}[/tex]then we have two values for l
[tex]\begin{gathered} l_1=\frac{165+\sqrt[]{3225}}{2}=110.9 \\ \\ l_2=\frac{165-\sqrt[]{3225}}{2}=54.1 \end{gathered}[/tex]we can take any value because both are positive and replace on any equation to find w
I will replace l=110.9 to find w
[tex]\begin{gathered} w\times l=6000 \\ w\times110.9=6000 \\ w=\frac{6000}{110.9} \\ \\ w=54.1 \end{gathered}[/tex]Finally the length and wifth of the other rectangle patks are
[tex]\begin{gathered} l=110.9 \\ w=54.1 \end{gathered}[/tex]meters
Cameron is playing 9 holes of golf. He needs to score a total of at most 14 over par on the last four holes tobeat his best golf score. On the last four holes, he scores 7 over par, 1 under par, 4 over par, and 1 underpar.Part 1 out of 3Enter and find the value of an expression that gives Cameron's score for 4 holes of golf.The expression isCameron's score is✓ CheckNext
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In golf:
[tex]\begin{gathered} \text{ par corresponds to 0 on the number line} \\ \text{ under par corresponds to negative numbers on the number line} \\ \text{over par corresponds to positive numbers on the number line} \end{gathered}[/tex]Hence, in this case,
[tex]\begin{gathered} 7\text{ over par corresponds to +7 on the number line} \\ 1\text{ under par corresponds to -1 on the number line} \\ 4\text{ over par corresponds to +4 on the number line} \end{gathered}[/tex]Hence, the value of an expression that gives Cameron's score for 4 holes of golf is given by
[tex]7-1+4-1=6+4-1=10-1=9[/tex]The expression is : 7 - 1 + 4 - 1
Camerons's score is: 9
I want to know the answer and steps please I would appreciate it.
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ANSWER
[tex]578.05yd^2[/tex]EXPLANATION
Given;
[tex]\begin{gathered} diameter(d)=8yd \\ radius(r)=\frac{d}{2}=\frac{8}{2}=4 \\ height(h)=19yd \end{gathered}[/tex]Recall, the formula for finding the surface area of a cylinder is;
[tex]A=2\pi rh+2\pi r^2[/tex]Substituting the values;
[tex]\begin{gathered} A=2\pi rh+2\pi r^2 \\ =2\times3.14\times4\times19+2\times3.14\times4^2 \\ =477.28+100.48 \\ =578.05 \end{gathered}[/tex]I need help with my math
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The pythagorean Theorem say:
[tex]h=\sqrt[]{l^2_1+l^2_2}[/tex]In this problem l1 and l2 will be a and b so:
[tex]h=\sqrt[]{14^2+18^2}[/tex]So finally we operate and we get:
[tex]\begin{gathered} h=\sqrt[]{196+324} \\ h=\sqrt[]{520} \\ h=22.8 \end{gathered}[/tex]find the slope (1, 2), (-3, 3)
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Given:
The points are (1, 2), (-3, 3).
The slope is calculated as,
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ (x_1,y_1)=\mleft(1,2\mright) \\ (x_2,y_2)=(-3,3) \\ m=\frac{3-2}{-3-1} \\ m=\frac{1}{-4} \\ m=-\frac{1}{4} \end{gathered}[/tex]Answer: slope = -1/4
To find the measure of
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You have the following expression:
(2z - 3) + (5z - 6) = 180
in order to solvet the previous expression for z, proceed as follow:
(2z - 3) + (5z - 6) = 180 cancel out parenthesis
2z - 3 + 5z - 6 = 180 simplify like terms left side
7z - 9 = 180 add 9 both sides
7z = 180 + 9
7z = 189 divide by 7 both sides
z = 189/7
z = 27
Hence, the value of z is 27
Next, replace the values of z into the expression for the measureof angle M:
Hence, the measure of angle M is 129°
I have no idea how to do this please help
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Answer:
Initial Value: 19,900
Value after 11 years: 7953
Explanation:
The initial value of the car is its value at t =0. Therefore, to find this initial value, we put in t = 0 to get
[tex]v(0)=19,900(0.92)^0[/tex][tex]\boxed{v\mleft(0\mright)=19,900}[/tex]Hence, the initial value is $19,900.
Now, to find the value after 11 years, we put t = 11 into the equation and get
[tex]v(11)=19,900(0.92)^{11}[/tex]which gives (rounded to the nearest dollar)
[tex]\boxed{v\mleft(11\mright)=7953}[/tex]which is our answer!
Hence, the value after 11 years is $7953.
If f(x) = 2x2 + x - 3, which equation can be used to determine the zeros of the function?
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Given the function:
[tex]f(x)=2x^2+x-3[/tex]to find the zeros of the function, we have to solve the equation f(x) = 0, this means the following:
[tex]\begin{gathered} f(x)=0 \\ \Rightarrow2x^2+x-3=0 \end{gathered}[/tex]solving for 'x', we get the zeros of the function, if there are any.
consider the function f(x) whose second derivative is f''(x)=4x+4sin(x). if f(0)=4 and f'(0)=2, what is f(3)?
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Select the correct phrase in the drop-down menu to complete the sentence for the first job down you have G(-1)____the answer can be either greater than h(-1) or equal to h(-1) or less than h(-1) . For the second drop down answer it has G(1)____ the answer can be either greater than h(1) or equal to h(1) or less than h(1)
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We are given the graph of a parabola represented by g(x) and the linear function h(x). To determine the value of g(-1) we go to the graph of the function and determine that the value is:
[tex]g(-1)=-2[/tex]To determine the value of h(-1) we replace the value of "x" in h(x) for -1:
[tex]\begin{gathered} h(-1)=-3(-1)+8 \\ h(-1)=3+8 \\ h(-1)=11 \end{gathered}[/tex]Therefore, since:
[tex]11>-2[/tex]we have that g(-1) is less than h(-1).
We do a similar procedure to determine g(1) from the graph:
[tex]g(1)=5[/tex]And we replace x = 1 in h(x) to get h(1):
[tex]\begin{gathered} h(1)=-3(1)+8 \\ h(1)=-3+8 \\ h(1)=5 \end{gathered}[/tex]Since we get the same value this means that g(1) is equal to h(1).
In circle o, a diameter has endpoints (-5,4) and(3, -6). What is the length of the diameter?Answer choices: a.) sqrt 10b.) 2sqrt 41c.) sqrt 2 d.) 2sqrt 2
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Diameter has endpoints (-5,4) and (3, -6).
the endpoints of the diameter = (-5,4) and (3,-6)
The length of diameter can be calculated by sqrt 10,11
Distance between two coordinates (-5,4) and (3,-6).
Distance formula is express as
[tex]\begin{gathered} \text{Distance = }\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^1^{}} \\ \text{Distance}=\sqrt[]{(3-(-5))+(-6+(4))} \\ \text{Diamter = }\sqrt[]{8-(-2)} \\ \text{Diameter = }\sqrt[]{10} \end{gathered}[/tex]Diameter = sqrt 10
Answer: Diameter = sqrt 10
What is the solution to the following equation?
3(x-4)-5 = x - 3A. x = 12B. x=3C. x=8D. x = 7
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Given:
3(x-4)-5 = x - 3
Required:
To calculate which option is correct
Explanation:
[tex]\begin{gathered} 3(x-4)-5=x-3 \\ \\ 3x-12-5=x-3 \\ \\ 3x-17=x-3 \\ \\ 3x-x=-3+17 \\ \\ 2x=14 \\ \\ x=\frac{14}{2} \\ \\ x=7 \end{gathered}[/tex]Required answer:
Option D (x=7)
Geometric Vectors in Cartesian Form
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Hello there. To talk about geometric vectors in cartesian form, we have to remember some properties about linear algebra.
Given two vectors u and v, we say they are written in cartesian coordinates if they have the following notation:
[tex]\begin{gathered} u=\langle u_1,\,u_2\rangle \\ \\ v=\langle v_1,\,v_2\rangle \end{gathered}[/tex]Of course, this notation is for vectors in two dimensions, so we say that
[tex]u,v\in\mathbb{V}^2[/tex]That is the vector space with two dimensions.
We can extend this to all the plane, considering the coordinates can take all values in the real numbers, hence
[tex]u,v\in\mathbb{R}^2[/tex]And finally extend this to n-dimensions, but in this case we cannot understand it geometrically since we can, at most, geometrically represent a vector up to three dimensions
[tex]u,v\in\mathbb{R}^n[/tex]Some properties about vectors:
They are associative, that means that
[tex](u+v)+w=u+(v+w)[/tex]We have also the distributive property
[tex](u+v)\cdot w=u\cdot w+v\cdot w[/tex]Whereas
[tex]\cdot\text{ is the scalar product operator}[/tex]It also holds for cross products and other kinds of products.
They are commutative
[tex]u+v=v+u[/tex]This holds for the scalar product:
[tex]u\cdot v=v\cdot u[/tex]but it doesn't for the cross product
[tex]u\times v=-v\times u[/tex]Now, we have the geometrical view of vectors.
Say we have a point (x, y) and we want to define a vector from this point.
So we plug the tail of the vector at the origin and its tip in the point, as follows:
We can define a vector from point to point as well, but we say that they are equipollent to a vector with its tail at the origin and has the same magnitude of the vector we found.
In higher dimensions, we have
In cartesian form, we can rewrite the vectors in the following notation:
The scalar product is defined as:
[tex]u\cdot v=\langle u_1,u_2\rangle\cdot\langle v_1,v_2\rangle=u_1v_1+u_2v_2[/tex]For higher dimensions, it holds that
[tex]u\cdot v=\langle u_1,u_2,\cdots,u_n\rangle\cdot\langle v_1,v_2,\cdots,v_n\rangle=\sum_{i=1}^nu_iv_i[/tex]These are the main properties about vectors.
function c is defined by the equation c(n)=50+4n. it gives the monthly cost in dollars of visitiing a gym as a function of the number of visits v. find the value of c (7). show your reasoning and explain what the value means in this situation
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The cost function is defined as
[tex]c(n)=50+4n[/tex]The value of c(7) can be determined by substituting 7 for n into the function.
Therefore, c(7) becomes;
[tex]\begin{gathered} c(7)=50+4(7) \\ c(7)=50+28 \\ c(7)=78 \end{gathered}[/tex]This means the cost of 7 visits to the gym is $78.
The cost function shows that there is an amount that doesnt change and that is 50. Then there is one that varies or changes with every visit, or n. That means as the value of n increases or decreases, the total amount also increases or decreases. When n equals zero, then the total cost becomes 50. This means when there is no visit to the gym, the cost still remains $50.
Therefore, n is a variable that can determine changes in the total cost.
Answer: c(7)=78
Step-by-step explanation:
C(n)=50+4n
c(7)=50+4(7)
=50+28
=78
C(7)=78
Round the value 23.731 g to three significant figures.Express your answer numerically using three significant figures.
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23.731 g rounded to three significant figures is 23.7 g.
Rule for significant figures:
All non-zero numbers are significant.
In 23.731, there are 5 non-zero numbers. So, there are 5 significant figures. To get number in 3 significant figures round the number at the first decimal place.
Now, the 23.731 g can be expressed in 3 significant figures as 23.7 g.
Now, use the angle measurement tool to measure the angles of each polygon. Do the angle measures agree with your results ?
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Solution: Answer is correct in three cases.
When we have a triangle, the sum of measures of angles are 180 degrees. In the three pictures of polygons, we have two triangles in each one. So, the sum of angles would be 360 degrees in all cases.
Case A: We have 85+80+100+x = 360. We isolate X,
X=360-85-80-100
X= 95 degrees.
Case B: We have 72+78+60+x = 360. We isolate X,
X=360-72-78-60
X= 150 degrees.
Case C: We have 90+90+30+x = 360. We isolate X,
X=360-90-90-30
X= 150 degrees.
Find the real part and the imaginary part of the following complex number. - 14 - 14/13
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determine the number light . determine the sign of each aggression below
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On the number line shown, a is less than 0, which means a is a negative number, while b is greater than 0 which means b is positive. Therefore;
(A) ab (that is a times b) = negative ab
(B) -(ab) = positive ab (observe that ab is already a negative value)
(C) 2a - b = negative (observe that 2a gives a negative result, hence subtracting 3 from a negative number will resut in a negative answer)
X 0 1 | 2 3 4 y 7 15 23 31 39
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step 1
Find the slope
we take the points
(0,7) and (1,15)
m=(15-7)/(1-0)
m=8/1
m=8
step 2
Find the equation in slope intercept form
y=mx+b
we have
m=8
b=7 -------> (0,7) is the y-intercept
substitute
y=8x+7Lynn lines the bottom of her first pan with aluminum foil. The area of the rectangular piece of foil is 11 1/4 square inches. It's length is 4 and 1/2 inches. what is the width of the foil
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The area of the rectangular foil is
[tex]\begin{gathered} \text{area}=11\frac{1}{4}inches^2=\frac{45}{4}inches^2 \\ \text{length}=4\frac{1}{2}inches=\frac{9}{2}inches^2 \\ \text{width =?} \\ \text{area}=\text{length}\times width \\ \frac{45}{4}=\frac{9}{2}w \\ \text{cross multiply} \\ 90=36w \\ w=\frac{90}{36}=\frac{30}{12}=\frac{10}{4}=\frac{5}{2}\text{ inches} \\ \end{gathered}[/tex]solve the following: a. 1/14 - 3 1/8 b. -7/2x^2y^2. + 5/2xy^2 + 3/x^2yc. -2 1/3 + 1 1/9
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First we analyze the denominators
2x^2y^2 is the greatest common factor for all three fractions, now we just divide each denominator with that factor and then multiply the answer by numerators
Then, the new fraction will be:
e)
how to compare fractions 15/8 and 12/7
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Answer:
To compare fractions 15/8 and 12/7
In order to comapare any two fractions, the denomiator of the two fractions must be same.
For the given fractions denominators are different.
To make the denomiator of the given fractions same, find the LCM of the denominators.
LCM of 8,7: 56
The fractions become,
[tex]\frac{15}{8}=\frac{15\times7}{8\times7}=\frac{105}{56}[/tex][tex]\frac{12}{7}=\frac{12\times8}{7\times8}=\frac{96}{56}[/tex]we get that,
[tex]\frac{105}{56}>\frac{96}{56}[/tex]we get,
[tex]\frac{15}{8}>\frac{12}{7}[/tex]15/8 is greater than 12/7.
Answer is: 15/8 is greater than 12/7
What is the difference between 63,209 and 8,846?
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From the question,
We are to find the difference between 63,209 and 8,846
Difference is given as
[tex]63,209-8,846[/tex]Therefore, we have
Therefore the difference between 63,209 and 8,846 is
54,363
In scientific notation, 0.00000729=?
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Answer:
7.29 x 10^-6
Step-by-step explanation:
the number must be between 1 and 9
then the 0 are 6
We put minus because we go to the left
7.29 x 10^ -6
which graph best repersents tge solution to the system of equations
y=x+4
y=3x-6
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The solution to the system of linear equation (x, y) are 5 and 9
Graph of Linear EquationsLinear equations, also known as first-order degree equations, where the highest power of the variable is one. When an equation has one variable, it is known as linear equations in one variable. If the linear equations contain two variables, then it is known as linear equations in two variables, and so on.
The solution of a linear equation in two variables is a pair of numbers, one for x and one for y which satisfies the equation. There are infinitely many solutions for a linear equation in two variables.
Therefore, every linear equation in two variables can be represented geometrically as a straight line in a coordinate plane. Points on the line are the solution of the equation. This why equations with degree one are called as linear equations. This representation of a linear equation is known as graphing of linear equations in two variables.
Using graph to solve this problem, the solution to the equations are (x, y) are 5 and 9 respectively.
Learn more on graph of linear equation here;
https://brainly.com/question/14323743
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Which of the equations below could be the equation of this parabola?5Vertex(0,0)10A. x = 2y2B. y= 2x2C. y = -2x2O D. x = -2y2
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Given:
Vertex = (0,0)
General vertex from of equation is:
[tex]y=a(x-h)^2+k[/tex][tex]\text{vertex = (h,k)}[/tex]So:
h = 0
k = 0
then equation is:
[tex]\begin{gathered} y=a(x-h)^2+k \\ h=0;k=0 \\ y=a(x-0)^2+0 \\ y=ax^2 \end{gathered}[/tex]Here value of "a" is negative because the graph is move downward . and all option give mode value is 2 then the equation of functuion:
[tex]y=-2x^2[/tex]