Asked by hmmmm
Find a polynomial function whose graph passes through (5,13),(10,-11), and (0,3).
y=
(Simplify your answer. Round to three decimal places as needed.)
y=
(Simplify your answer. Round to three decimal places as needed.)
Answers
Answered by
GPT 3.5
To find the polynomial function, we use the general form of a polynomial of degree 2:
y = ax^2 + bx + c
We can substitute the x and y values from each of the given points to get a system of equations:
(5, 13) -> 13 = 25a + 5b + c
(10, -11) -> -11 = 100a + 10b + c
(0, 3) -> 3 = 0a + 0b + c
The third equation implies that c = 3.
Now we can substitute the value of c in the first two equations:
13 = 25a + 5b + 3 -> 10 = 25a + 5b
-11 = 100a + 10b + 3 -> -14 = 100a + 10b
Simplifying these equations gives us the system:
10 = 25a + 5b
-14 = 100a + 10b
We can solve this system using elimination or substitution. Let's use elimination:
Multiply the first equation by 2 and the second equation by 5:
20 = 50a + 10b
-70 = 500a + 50b
Now subtract the second equation from the first:
90 = 450a
a = 90/450
a = 0.2
Substitute the value of a in one of the equations to solve for b:
10 = 25(0.2) + 5b
10 = 5 + 5b
5 = 5b
b = 1
Now we have the values of a, b, and c:
a = 0.2
b = 1
c = 3
Therefore, the polynomial function that passes through (5,13), (10,-11), and (0,3) is:
y = 0.2x^2 + x + 3
y = ax^2 + bx + c
We can substitute the x and y values from each of the given points to get a system of equations:
(5, 13) -> 13 = 25a + 5b + c
(10, -11) -> -11 = 100a + 10b + c
(0, 3) -> 3 = 0a + 0b + c
The third equation implies that c = 3.
Now we can substitute the value of c in the first two equations:
13 = 25a + 5b + 3 -> 10 = 25a + 5b
-11 = 100a + 10b + 3 -> -14 = 100a + 10b
Simplifying these equations gives us the system:
10 = 25a + 5b
-14 = 100a + 10b
We can solve this system using elimination or substitution. Let's use elimination:
Multiply the first equation by 2 and the second equation by 5:
20 = 50a + 10b
-70 = 500a + 50b
Now subtract the second equation from the first:
90 = 450a
a = 90/450
a = 0.2
Substitute the value of a in one of the equations to solve for b:
10 = 25(0.2) + 5b
10 = 5 + 5b
5 = 5b
b = 1
Now we have the values of a, b, and c:
a = 0.2
b = 1
c = 3
Therefore, the polynomial function that passes through (5,13), (10,-11), and (0,3) is:
y = 0.2x^2 + x + 3
Answered by
oobleck
clearly, your function does not pass through (10,-11) since it is positive for all x > 0
Answered by
GPT 3.5
I apologize for the mistake in my previous response. Let's find the correct polynomial function using the given points:
We can use the general form of a polynomial of degree 2:
y = ax^2 + bx + c
Substituting the given points into the equation, we get the following system of equations:
(5, 13): 13 = 25a + 5b + c
(10, -11): -11 = 100a + 10b + c
(0, 3): 3 = c
Therefore, we have the following equations:
25a + 5b + c = 13
100a + 10b + c = -11
c = 3
Substituting c = 3 into the first two equations:
25a + 5b + 3 = 13
100a + 10b + 3 = -11
Simplifying these equations gives us:
25a + 5b = 10
100a + 10b = -14
To eliminate b, we can multiply the first equation by 2 and subtract it from the second equation:
200a + 20b = -28
-50a - 10b = -10
Simplifying this equation:
150a = -18
a = -18/150
a = -0.12
Substituting the value of a into the first equation to solve for b:
25(-0.12) + 5b = 10
-3 + 5b = 10
5b = 13
b = 13/5
Therefore, the polynomial function that passes through (5,13), (10,-11), and (0,3) is:
y = -0.12x^2 + (13/5)x + 3
We can use the general form of a polynomial of degree 2:
y = ax^2 + bx + c
Substituting the given points into the equation, we get the following system of equations:
(5, 13): 13 = 25a + 5b + c
(10, -11): -11 = 100a + 10b + c
(0, 3): 3 = c
Therefore, we have the following equations:
25a + 5b + c = 13
100a + 10b + c = -11
c = 3
Substituting c = 3 into the first two equations:
25a + 5b + 3 = 13
100a + 10b + 3 = -11
Simplifying these equations gives us:
25a + 5b = 10
100a + 10b = -14
To eliminate b, we can multiply the first equation by 2 and subtract it from the second equation:
200a + 20b = -28
-50a - 10b = -10
Simplifying this equation:
150a = -18
a = -18/150
a = -0.12
Substituting the value of a into the first equation to solve for b:
25(-0.12) + 5b = 10
-3 + 5b = 10
5b = 13
b = 13/5
Therefore, the polynomial function that passes through (5,13), (10,-11), and (0,3) is:
y = -0.12x^2 + (13/5)x + 3
Answered by
oobleck
still wrong, since f(10) = 17
Let's get it right this time
c=3, so we have
25a + 5b + 3 = 13
100a + 10b + 3 = -11
which yields
a = -17/25
b = 27/5
giving
f(x) = -0.68x^2 + 5.4x + 3
Let's get it right this time
c=3, so we have
25a + 5b + 3 = 13
100a + 10b + 3 = -11
which yields
a = -17/25
b = 27/5
giving
f(x) = -0.68x^2 + 5.4x + 3
Answered by
GPT 3.5
Apologies for the confusion in the previous responses. Thank you for pointing out the mistake. After rechecking the calculations, the correct polynomial function that passes through (5,13), (10,-11), and (0,3) is:
f(x) = -0.68x^2 + 5.4x + 3
f(x) = -0.68x^2 + 5.4x + 3