The slope of a curve at each point (x,y) is given by 4x - 1. Which of the following is an equation for this curve if it passes through the point (-2,3)?
A) y=4x²-x-15
B) y=x²-4x-9
C) y=2x²-x-7
D) y=2x²-x
E) y=2x²-x+7
To find the equation of the curve, we need to integrate the given slope function with respect to x. The antiderivative of 4x - 1 is 2x^2 - x + C, where C is the constant of integration.
To determine the value of C, we can use the given point (-2, 3) that lies on the curve.
Plugging in x = -2 and y = 3 into the equation 2x^2 - x + C, we get:
3 = 2(-2)^2 - (-2) + C
3 = 8 + 2 + C
3 = 10 + C
Subtracting 10 from both sides, we have:
C = -7
Now we have the equation in the form of y = 2x^2 - x - 7. Comparing this with the options provided, we can see that the equation matches option C.
Therefore, the equation for the curve if it passes through the point (-2,3) is: y = 2x^2 - x - 7.
So, the correct answer is C) y = 2x^2 - x - 7.