Determine an expression, in simplified form, for the slope of the secant PQ.
Given: P(1,3) , Q(1+h, f(1+h)) , where f(x) = 3x^2
PQ=Y2-Y1/X2-X1
I keep getting wrong answer, arghhh.
To find the slope of the secant PQ, we can use the formula:
slope = (Y2 - Y1) / (X2 - X1)
In this case, the coordinates of P are (1,3), and the coordinates of Q are (1+h, f(1+h)).
So, substituting the values into the slope formula, we get:
slope = [f(1+h) - 3] / [1+h - 1]
Now let's substitute the given function f(x) = 3x^2 into the equation:
slope = [3(1+h)^2 - 3] / h
Expanding (1+h)^2, we get:
slope = [3(1 + 2h + h^2) - 3] / h
Now simplify further:
slope = [3 + 6h + 3h^2 - 3] / h
= (6h + 3h^2) / h
= 6 + 3h
So, the expression for the slope of the secant PQ is 6 + 3h in simplified form.
To determine the slope of the secant PQ, we need to use the slope formula, which is:
slope (m) = (Y2 - Y1) / (X2 - X1)
Given that P(1,3) and Q(1+h, f(1+h)), we can substitute the coordinates into the formula to find the slope.
First, let's find the values for Y1, Y2, X1, and X2:
Y1 = 3 (from point P)
Y2 = f(1+h) = 3(1+h)^2 (from point Q and the given equation f(x) = 3x^2)
X1 = 1 (from point P)
X2 = 1+h (from point Q)
Now we can substitute these values into the slope formula:
slope (m) = (Y2 - Y1) / (X2 - X1)
= [3(1+h)^2 - 3] / (1+h - 1)
= [3(1+2h+h^2) - 3] / h
To simplify further, we can expand the square term:
slope (m) = [3(1+2h+h^2) - 3] / h
= (3 + 6h + 3h^2 - 3) / h
= (6h + 3h^2) / h
= 6 + 3h
So, the expression in simplified form for the slope of the secant PQ is 6 + 3h.
f(1+h) = 2(1+h)^2 = 3(1 + 2h + h^2)
= 3 + 6h + 3h^2
f(1) = 3
slope of secant = (3 + 6h - 3h^2 - 3)/(1+h - 1)
= (6h - 3h^2)/h
= 6 - 3h