Determine the values of the coefficients a, b, c, d, and e so that the following f(x) is a natural cubic spline on [0, 2]: f(x) = (1 + x − ax2 + bx3                                                 , x ∈ [0, 1]            c + d(x − 1) + e(x − 2)2 + (x

Q: Which function represents a transformation of the parent quadratic function and 2 units down? a. fx)…

A: Here we have to transform the parent qau8dratic function 6 units to the left and 2 units down.

Q: 3. Find the roots of ( x2-1) ( x2 + 3x – 2)=0

A:

Q: The Range of the quadratic function f(x) = -x² + 15x + 45 is O (0,8] О -45, 8 O [-8, 0) O [45, 0) O…

A: By sketching the graph

Q: 5. Find a polynomial of degree 3 whose zeros are 4, 1,-2 and y intercept 16.

A: Since given that the polynomial has degree 3 and its zeros are 4, 1, and -2.Then the polynomial can…

Q: 1) Sketch a cubic polynomial whose zeros are -1, 2, and 9

A:

Q: Find all roots of z(z² + 5x – 25) = 125 and their multiplicity: I = -5 with multiplicity 3 I = -125,…

A: Here we find all roots of    x (x2+5x-25) =125and their multiplicity

Q: Let (-1,(-1) = 2), (0, ƒ(0) = 5) and (1, f(1) = 6). Which one gives quadratic spline functions?…

A: Given, -1,f(-1)=2,(0,f(0)=5) and 1,f(1)=6

Q: Show that (x? + y²)² = (x² – y²)² + (2xy)? for all real numbers x and y.

A: We have to prove algebraically that x2+y22=x2-y22+2xy2 Simplifying both the sides , we get Here ,…

Q: 2, and t = 3k – 2, find an expression for y = f(k). b) Express y as a function of k if y = 2x + 5

A: In the given question, the function yx is solved for yk. Given, yx=3x-2xt=3t+2tk=3k-2