y= f(x)=x^2-x-1 y= f(x)=x^3-x^2-2x+1 Write Python codes to plot the given function using Matplotlib library. This will give an idea where the roots of the given function are located.

y= f(x)=x^2-x-1 y = = f(x)=x^3-x^2-2x+1 Implement Bisection Method to find all the possible roots of these 2 given functions and verify it with built-in functions scipy.optimize.root() in the SciPy library. follow this given template # Root Finding Method import math import numpy as np import scipy as sp import matplotlib.pyplot as plt def plot_function(func, a, b): """ This function plot the graph of the input func within the given interval [a,b). """ # Your code goes here def bisection_method(func, a, b, tol=1e-6, max_iter=100): """ Bisection method to find the root of a function within a given interval. Parameters: - func: The function for which the root is to be found. - a, b: Interval [a, b] within which the root is searched for. - tol: Tolerance level for checking convergence of the method. - max_iter: Maximum number of iterations. Returns: - root: Approximation of the root. Example -------- >>> fun = lambda x: x**2 - x - 1 >>> root = bisection_method(fun, 1, 2, max_iter=20) """ # Check if the interval is valid (signs of f(a) and f(b) are different) # Your code goes here # Main loop starts here iter_count = 1 while iter_count <= max_iter: # your code goes here iter_count += 1 print("Warning! Exceeded the maximum number of iterations.") return root # Example usage: if __name__ == "__main__": # Define the function for which the root is to be found func = lambda x: x**2 - x - 1 # First Function # Uncomment the below line to use the Second Function # func = lambda x: x**3 - x**2 - 2*x + 1 # Second Function # Call plot_function to plot graph of the function # Your code goes here # Set the interval [a, b] for the search a_1 = 0; b_1 = 0; # For first root (change the values as required) a_2 = 0; b_2 = 0; # For second root (change the values as required) # Call the bisection method our_root_1 = # your code goes here our_root_2 = # your code goes here # Call SciPy method root, which we consider as a reference method. x0 = (a_1 + b_1)/2 sp_result_1 = sp.optimize.root(func, x0) sp_root_1 = sp_result_1.x.item() x0 = (a_2 + b_2)/2 sp_result_2 = sp.optimize.root(func, x0) sp_root_2 = sp_result_2.x.item() # Print the result print("1st root found by Bisection Method = {:0.8f}.".format(our_root_1)) print("1st root found by SciPy = {:0.8f}".format(sp_root_1)) print("2nd root found by Bisection Method = {:0.8f}.".format(our_root_2)) print("2nd root found by SciPy = {:0.8f}".format(sp_root_2))

Implement Newton-Raphson method to find all the possible roots of the given function and verify it with built-in functions scipy.optimize.root() in the SciPy library. Given Functions are: y= f(x)=x^2-x-1 y= f(x)=x^3-x^2-2x+1 follow the template please : # Root Finding Method import math import numpy as np import scipy as sp import matplotlib.pyplot as plt def plot_function(func, a, b): """ This function plot the graph of the input func within the given interval [a,b). """ # Your code goes here def newton_method(func, grad, x0, tol=1e-6, max_iter=100): '''Approximate solution of f(x)=0 by Newton-Raphson's method. Parameters ---------- func : function Function value for which we are searching for a solution f(x)=0, grad: function Gradient value of function f(x) x0 : number Initial guess for a solution f(x)=0. tol : number Stopping criteria is abs(f(x)) < tol. max_iter : integer Maximum number of iterations of Newton's method. Returns ------- xn : root Example -------- >>> fun = lambda x: x**2 - x - 1 >>> grad = lambda x: 2*x - 1 >>> root = newton_method(fun, grad, 1, max_iter=20) ''' # Main Loop starts here iter_count = 1 while iter_count <= max_iter: # Your code goes here iter_count += 1 print("Warning! Exceeded the maximum number of iterations.") return root # Main Driver Function: if __name__ == "__main__": # Define the 1st Function for which the root is to be found func = lambda x: x**2 - x - 1 # Define the gradient of the Function grad = lambda x: 2*x -1 # Uncomment the next two lines to use the 2nd Function #func = lambda x: x**3 - x**2 - 2*x + 1 #grad = lambda x: 3*x**2 - 2*x -2 # Call plot_function to plot graph of the function # Your code goes here x0 = 0 # Initial guess for 1st (change the value as required) # Call the Newton's method for 1st root our_root_1 = # Your code goes here # Call SciPy method (reference method) for 1st root sp_result_1 = sp.optimize.root(func, x0) sp_root_1 = sp_result_1.x.item() # Call the Newton's method for 2nd root x0 = 0 # Initial guess for 2nd root (change the value as required) our_root_2 = # Your code goes here # Call SciPy method (reference method) for 2nd root sp_result_2 = sp.optimize.root(func, x0) sp_root_2 = sp_result_2.x.item() # Print the result print("1st root found by Newton's Method = {:0.8f}.".format(our_root_1)) print("1st root found by SciPy = {:0.8f}".format(sp_root_1)) print("2nd root found by Newton's Method = {:0.8f}.".format(our_root_2)) print("2nd root found by SciPy = {:0.8f}".format(sp_root_2))

Graph the equation y, equals, minus, x, squared, minus, 4, x, minus, 3y=−x 2 −4x−3 on the accompanying set of axes. You must plot 5 points including the roots and the vertex.Click to plot points. Click points to delete them.-10-9-8-7-6-5-4-3-2-112345678910-10-9-8-7-6-5-4-3-2-112345678910xyAnswerAttempt 1 out of 2You must answer all questions above in order to submit.

Wrie a python program to use a function that computes the roots of a quadratic equation

The graph of a function f is given in the figure.A curve is shown on the x y coordinate plane. It begins at the point (−2, −1), goes up and to the right, passes through the approximate point (−1, −0.2), and passes through the negative x-axis at the approximate point (−0.8, 0). It then continues up and right, passes through the positive y-axis at the point (0, 1), and reaches a high point at (1, 3). It then goes down and right, passes through the points (2, 2) and (3, 1), and ends at the approximate point (4, 0.5).(a)Find the value of f(1).(b)Estimate the value of f(−1).(c)For what values of x is f(x) = 1? (Enter your answers as a comma-separated list.) (d)Estimate the value of x such that f(x) = 0.x = (e)State the domain and range of f. (Enter your answers in interval notation.)domain range (f)On what interval is f increasing? (Enter your answer using interval notation.)