Substitute (0, 1) into the equation.
y = ax² + bx + c
1 = a(0)^2 + b(0) + c
1 = c
Substitute (1, 0) into the equation.
y = ax² + bx + c
0 = a(1)^2 + b(1) + c
0 = a + b + c
The derivative is 1 because the slope is 1.
We can solve using the basic differentiation rules.
~Sum and difference rule.
= d/dx[f(x) ± g(x)]
= d/dx[f(x)] ± d/dx[g(x)]
~Constant multiple rule.
= d/dx[cf(x)] = cf(x)
~Constant rule.
= d/dx(c) = 0
~Power rule.
y = d/dx(x^n) = nx^(n-1)
y = dy/dx
y = d/dx(ax^2) + d/dx(bx) + d/dx(c)
y = a*(d/dx)(x^2) + b*(d/dx)(x) + 0
y = 2ax + b
Now plug; y = 1 and x = 1
1 = 2a(1) + b
1 = 2a + b
Now, we have a system.
c = 1
a + b + c = 0
2a + b = 1
~Simplify since c = 1
a + b + 1 = 0
2a + b = 1
Now, we only have two equations in the system.
Solve both equations for b.
a + b + 1 = 0
b + 1 = -a
b = -a - 1
2a + b = 1
b = 1 - 2a
Solve for a.
-a - 1 = 1 - 2a
-a = 2 - 2a
a = 2
Plug a in either of both equations and solve for b.
a + b + 1 = 0
2 + b + 1 = 0
3 + b = 0
b = -3
Now, we have all variables.
a = 2
b = -3
c = 1
Finally, substitute all the variables into the formula.
The equation of the variable is [ y = 2x² - 3x + 1 ]
Best of Luck!