+495 Okay, this one is tough. I'm not sure it's even possible, but the experts on here are more knowledgeable than me, so let's get to it.
I need a function that goes through these points:
(1, 1)
(2, 1.6)
(3, 2.4)
(4, 3)
(5, 4)
(6, 5)
(7, 6.4)
(8, 8)
That's it! That's all I need. It might be super easy for a graphing expert but maybe not.
This is the best I've been able to come up with but it's not even close:
\(y = \frac{\left (\left ({x-1}\right )^{\frac{4}{3}} \right )}{2} +1\)
Any help would be SUPER appreciated!
EDIT: you can use any and as many exponents as you need and you can use any tricks to make it happen!
+26367 I need a function that goes through these points:
(1, 1)
(2, 1.6)
(3, 2.4)
(4, 3)
(5, 4)
(6, 5)
(7, 6.4)
(8, 8)
Function: \(\begin{array}{rcll} y &=& ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h \\ \end{array} \)
We have eight equations:
\(\begin{array}{rcll} 1 &=& a\cdot (1)^7 + b\cdot (1)^6 + c\cdot (1)^5 + d\cdot (1)^4 + e\cdot (1)^3 + f\cdot (1)^2 + g\cdot (1) + h \\ 1.6 &=& a\cdot (2)^7 + b\cdot (2)^6 + c\cdot (2)^5 + d\cdot (2)^4 + e\cdot (2)^3 + f\cdot (2)^2 + g\cdot (2) + h \\ 2.4 &=& a\cdot (3)^7 + b\cdot (3)^6 + c\cdot (3)^5 + d\cdot (3)^4 + e\cdot (3)^3 + f\cdot (3)^2 + g\cdot (3) + h \\ 3 &=& a\cdot (4)^7 + b\cdot (4)^6 + c\cdot (4)^5 + d\cdot (4)^4 + e\cdot (4)^3 + f\cdot (4)^2 + g\cdot (4) + h \\ 4 &=& a\cdot (5)^7 + b\cdot (5)^6 + c\cdot (5)^5 + d\cdot (5)^4 + e\cdot (5)^3 + f\cdot (5)^2 + g\cdot (5) + h \\ 5 &=& a\cdot (6)^7 + b\cdot (6)^6 + c\cdot (6)^5 + d\cdot (6)^4 + e\cdot (6)^3 + f\cdot (6)^2 + g\cdot (6) + h \\ 6.4 &=& a\cdot (7)^7 + b\cdot (7)^6 + c\cdot (7)^5 + d\cdot (7)^4 + e\cdot (7)^3 + f\cdot (7)^2 + g\cdot (7) + h \\ 8 &=& a\cdot (8)^7 + b\cdot (8)^6 + c\cdot (8)^5 + d\cdot (8)^4 + e\cdot (8)^3 + f\cdot (8)^2 + g\cdot (8) + h \\ \end{array}\)
We need the inverse Matrix of:
( 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, {nl} 128.000000, 64.000000, 32.000000, 16.000000, 8.000000, 4.000000, 2.000000, 1.000000, {nl} 2187.000000, 729.000000, 243.000000, 81.000000, 27.000000, 9.000000, 3.000000, 1.000000, {nl} 16384.000000, 4096.000000, 1024.000000, 256.000000, 64.000000, 16.000000, 4.000000, 1.000000, {nl} 78125.000000, 15625.000000, 3125.000000, 625.000000, 125.000000, 25.000000, 5.000000, 1.000000, {nl} 279936.000000, 46656.000000, 7776.000000, 1296.000000, 216.000000, 36.000000, 6.000000, 1.000000, {nl} 823543.000000, 117649.000000, 16807.000000, 2401.000000, 343.000000, 49.000000, 7.000000, 1.000000, {nl} 2097152.000000, 262144.000000, 32768.000000, 4096.000000, 512.000000, 64.000000, 8.000000, 1.000000 )
The inverse Matrix is:
( -0.000198, 0.001389, -0.004167, 0.006944, -0.006944, 0.004167, -0.001389, 0.000198,
0.006944, -0.047222, 0.137500, -0.222222, 0.215278, -0.125000, 0.040278, -0.005556,
-0.101389, 0.663889, -1.862500, 2.902778, -2.715278, 1.525000, -0.476389, 0.063889,
0.798611, -4.972222, 13.312500, -19.888889, 17.923611, -9.750000, 2.965278, -0.388889,
-3.655556, 21.234722, -53.600000, 76.340278, -66.277778, 35.037500, -10.422222, 1.343056,
9.694444, -50.980556, 119.550000, -161.888889, 135.861111, -70.125000, 20.494444, -2.605556,
-13.742857, 62.100000, -133.533333, 172.750000, -141.000000, 71.433333, -20.600000, 2.592857,
8.000000, -28.000000, 56.000000, -70.000000, 56.000000, -28.000000, 8.000000, -1.000000 )
The coefficients (a,b,c,d,e,f,g,h) are:
a = -0.0013888889 {nl} b = 0.0441666667 {nl} c = -0.5747222222 {nl} d = 3.9375000000 {nl} e = -15.1805555556 {nl} f = 32.5183333333 {nl} g = -34.5433333333 {nl} h = 14.8000000000
The function is:
\(\small{ \begin{array}{rcll} y &=& -0.0013888889 \cdot x^7 + 0.0441666667\cdot x^6 -0.5747222222\cdot x^5 +\\ && +3.9375 \cdot x^4 -15.1805555556 \cdot x^3 + 32.5183333333\cdot x^2 -34.5433333333\cdot x + 14.8 \\ \end{array} }\)
or
\(\small{ \begin{array}{rcll} y &=& -\frac{1}{720} \cdot x^7 + \frac{53}{1200}\cdot x^6 -\frac{2069}{3600}\cdot x^5 + \frac{63}{16} \cdot x^4 -\frac{1093}{72} \cdot x^3 + 32.518\overline{3}\cdot x^2 -34.54\overline{3}\cdot x + \frac{74}{5} \\ \end{array} }\)
+128460 LambLamb....this is a VERY difficult thing to do !!!!.......It can be managed with something called "LaGrange Polynomial Interpolation,' but this process is tedious even for a couple of points, much less eight of them....!!!!
I tried using WolframAlpha [ a very sophistacated online tool ].....it would handle up to the firstt five points, but after that, it apparently developed "sensory overload" (LOL!!!!)
Perhaps some better mathematicians than me [ Alan, Bertie, Heureka, etc.] know how to do this using other techniques/resources......???
Sorry !!!!!!
+26367 I need a function that goes through these points:
(1, 1)
(2, 1.6)
(3, 2.4)
(4, 3)
(5, 4)
(6, 5)
(7, 6.4)
(8, 8)
Function: \(\begin{array}{rcll} y &=& ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h \\ \end{array} \)
We have eight equations:
\(\begin{array}{rcll} 1 &=& a\cdot (1)^7 + b\cdot (1)^6 + c\cdot (1)^5 + d\cdot (1)^4 + e\cdot (1)^3 + f\cdot (1)^2 + g\cdot (1) + h \\ 1.6 &=& a\cdot (2)^7 + b\cdot (2)^6 + c\cdot (2)^5 + d\cdot (2)^4 + e\cdot (2)^3 + f\cdot (2)^2 + g\cdot (2) + h \\ 2.4 &=& a\cdot (3)^7 + b\cdot (3)^6 + c\cdot (3)^5 + d\cdot (3)^4 + e\cdot (3)^3 + f\cdot (3)^2 + g\cdot (3) + h \\ 3 &=& a\cdot (4)^7 + b\cdot (4)^6 + c\cdot (4)^5 + d\cdot (4)^4 + e\cdot (4)^3 + f\cdot (4)^2 + g\cdot (4) + h \\ 4 &=& a\cdot (5)^7 + b\cdot (5)^6 + c\cdot (5)^5 + d\cdot (5)^4 + e\cdot (5)^3 + f\cdot (5)^2 + g\cdot (5) + h \\ 5 &=& a\cdot (6)^7 + b\cdot (6)^6 + c\cdot (6)^5 + d\cdot (6)^4 + e\cdot (6)^3 + f\cdot (6)^2 + g\cdot (6) + h \\ 6.4 &=& a\cdot (7)^7 + b\cdot (7)^6 + c\cdot (7)^5 + d\cdot (7)^4 + e\cdot (7)^3 + f\cdot (7)^2 + g\cdot (7) + h \\ 8 &=& a\cdot (8)^7 + b\cdot (8)^6 + c\cdot (8)^5 + d\cdot (8)^4 + e\cdot (8)^3 + f\cdot (8)^2 + g\cdot (8) + h \\ \end{array}\)
We need the inverse Matrix of:
( 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, 1.000000, {nl} 128.000000, 64.000000, 32.000000, 16.000000, 8.000000, 4.000000, 2.000000, 1.000000, {nl} 2187.000000, 729.000000, 243.000000, 81.000000, 27.000000, 9.000000, 3.000000, 1.000000, {nl} 16384.000000, 4096.000000, 1024.000000, 256.000000, 64.000000, 16.000000, 4.000000, 1.000000, {nl} 78125.000000, 15625.000000, 3125.000000, 625.000000, 125.000000, 25.000000, 5.000000, 1.000000, {nl} 279936.000000, 46656.000000, 7776.000000, 1296.000000, 216.000000, 36.000000, 6.000000, 1.000000, {nl} 823543.000000, 117649.000000, 16807.000000, 2401.000000, 343.000000, 49.000000, 7.000000, 1.000000, {nl} 2097152.000000, 262144.000000, 32768.000000, 4096.000000, 512.000000, 64.000000, 8.000000, 1.000000 )
The inverse Matrix is:
( -0.000198, 0.001389, -0.004167, 0.006944, -0.006944, 0.004167, -0.001389, 0.000198,
0.006944, -0.047222, 0.137500, -0.222222, 0.215278, -0.125000, 0.040278, -0.005556,
-0.101389, 0.663889, -1.862500, 2.902778, -2.715278, 1.525000, -0.476389, 0.063889,
0.798611, -4.972222, 13.312500, -19.888889, 17.923611, -9.750000, 2.965278, -0.388889,
-3.655556, 21.234722, -53.600000, 76.340278, -66.277778, 35.037500, -10.422222, 1.343056,
9.694444, -50.980556, 119.550000, -161.888889, 135.861111, -70.125000, 20.494444, -2.605556,
-13.742857, 62.100000, -133.533333, 172.750000, -141.000000, 71.433333, -20.600000, 2.592857,
8.000000, -28.000000, 56.000000, -70.000000, 56.000000, -28.000000, 8.000000, -1.000000 )
The coefficients (a,b,c,d,e,f,g,h) are:
a = -0.0013888889 {nl} b = 0.0441666667 {nl} c = -0.5747222222 {nl} d = 3.9375000000 {nl} e = -15.1805555556 {nl} f = 32.5183333333 {nl} g = -34.5433333333 {nl} h = 14.8000000000
The function is:
\(\small{ \begin{array}{rcll} y &=& -0.0013888889 \cdot x^7 + 0.0441666667\cdot x^6 -0.5747222222\cdot x^5 +\\ && +3.9375 \cdot x^4 -15.1805555556 \cdot x^3 + 32.5183333333\cdot x^2 -34.5433333333\cdot x + 14.8 \\ \end{array} }\)
or
\(\small{ \begin{array}{rcll} y &=& -\frac{1}{720} \cdot x^7 + \frac{53}{1200}\cdot x^6 -\frac{2069}{3600}\cdot x^5 + \frac{63}{16} \cdot x^4 -\frac{1093}{72} \cdot x^3 + 32.518\overline{3}\cdot x^2 -34.54\overline{3}\cdot x + \frac{74}{5} \\ \end{array} }\)
