Consider de quadratic function y=ax²+bx+c:
a. Use the mathematical and graphical analyses to determine the equation of the above function if the parabola cross the line y=2x at two points with the x-coordinates 0 and 10. Hint: there is more than one solution.
b. determine the domain and range of the variables
c. how your solutions for a and b be modified if y=ax²+bx+c is a revenue function
I did the following work
a. The parabola crosses line y=2x at 2 points with the x-coordinates of 0 and 10.
Replacing x-coordinate of 0: y=2x y=2(0) = 0 point (0,0)
Replacing x-coordinate of 10: y=2x y=2(10) = 20 point (10,20)
Therefore, the parabola crosses line y=2x at at points (0,0) and (10,20). Replacing both points in the quadratic function y=ax2+bx+c : finally the function is y = ax2 + (2-10a) x (a > 0, a < 0)
b. When a > 0, the vertex is a minimum:
Finding the vertex vertex: x= -b/ 2a= - (2-10a)/2a = 10a-2/2a= 5-1/a
Replacing x= 5-1/a in the formula to find y
y= a(5-1/a)2 + (2-10a)(5-1/a) = -25a-1/a+10
D: {x| all real numbers} & R: {y| y≥5-1/a}
When a < 0, the vertex is a maximum:
D: {x| all real numbers} & R: {y| y≤5-1/a}