Could you please help me with this question.
Determine an expression, in simplified form, for the slope of the secant PQ.
a.) P(1,3), Q(1+h,f(1+h)), where f(x)=3x^2
To determine the slope of the secant PQ, we need to find the change in y-values divided by the change in x-values between the points P and Q.
The y-coordinate of point P is 3, and the y-coordinate of point Q is f(1+h).
To find the y-coordinate of Q, we substitute the value of x = 1+h into the function f(x)=3x^2:
f(1+h) = 3(1 + h)^2
= 3(1 + 2h + h^2)
= 3 + 6h + 3h^2
So, the y-coordinate of point Q is 3 + 6h + 3h^2.
Now, let's determine the change in y-values:
Δy = (y-coordinate of Q) - (y-coordinate of P)
= (3 + 6h + 3h^2) - 3
= 6h + 3h^2
The change in x-values is h, as x remains constant between P and Q.
Therefore, the slope of the secant PQ is given by the expression:
slope = Δy / Δx
= (6h + 3h^2) / h
= 6 + 3h
Hence, the expression for the slope of the secant PQ is 6 + 3h.