Consider the following points.

Question

Consider the following points.

(−1, 2), (0, 0), (1, 1), (4, 58)

(a) Determine the polynomial function of least degree whose graph passes through the given points.

Answer 1

find polynomial function of least degree whose graph passes through

(-1,2), (0,0), (1,1), (4,58)

it can't be more than a cubic polynomial.

they aren't on a straight line so rule out linear

that leaves either a degree 2 or 3 polynomial

plot the 4 points. visually it looks like a possible parabola unpward opening

y = ax^2+bx + c

plug in (0,0) to get 0 = 0+0+c, c=0

then

y = ax^2 +bx

plug in each of 2 other points, to get y=3x^2/2 -x/2

but the point (4,58) won't fit,

so move on to a cubic

guaranteed there is a cubic polynomial

y=ax^3+ bx^2+cx + d

plug in the origin, d= 0

y = ax^3+bx^2 + cx

plug in (1,1)

1 = a+b+c

plug in the other points to get total 3 equations 3 unknowns

solve by substitution, elimination

y = (3/5)x^3 +3x^2 -(11/10))x

is the cubic polynomial through the 4 given points

or

y =.6x^3+1.5x^2 -1.1x

check by plugging in each of the 4 points and see if they satisfy the cubic polynomial function

Answer 2

p(x) = 0.6x³ + 1.5x² -1.1x

Answer 3

Try y = Ax³ + Bx² + Cx + D

Since (0,0) is on the curve, D = 0

So, we have the following system of 3 equations in 3 unknowns:

A + B + C = 1

-A + B - C = 2

64A + 16B + 4C = 58

Adding the first 2 equations yields 2B = 3. So, B = 3/2.

Using equation 1 and equation 3, we get:

A + C = -1/2

64A + 4C = 34

-64A - 64C = 32

64A + 4C = 34

-60C = 66

So, C = -11/10

A - 11/10 = -1/2

A = 3/5

So, y = (3/5)x³ + (3/2)x² - (11/10)x

Answer 4

A parabola is symmetrical and there is no symmetry with the given points so we can skip to a third degree polynomial:

y = ax³ + bx² + cx + d

Plugging in the 4 dat points would result in the following 4 equations:

1) 2 = -a + b - c + d

2) 0 = d

3) 1 = a + b + c + d

4) 58 = 64a + 16b + 4c + d

Knowing that d = 0, we can simplify the other 3 equations into:

a) -a + b - c = 2

b) a + b + c = 1

c) 64a + 16b + 4c = 58 or, dividing by 2, 32a + 8b + 2c = 29

Adding equation a) and b) together yields 2b = 3 or b = 1.5

Using equations a) and c) and plugging in b = 1.5 results in:

-a - c = 0.5

32a + 2c = 17

Multiplying -a - c = 0.5 by 2 gives -2a - 2c = 1 and then adding this to 32a + 2c = 17 gives 30a = 18 or a = 18/30 = 3/5

Then using -a - c = 0.5 and plugging in a = 9/15 we get c = -1.1

Plug those values into y = ax³ + bx² + cx + d for the answer