#### furor celtica

make a generalisation about the gradient at any point on the graph of y=x^2 + c, where c is any real number

my textbook gives no answer for this, so i'd just like to make sure that the correct answer is that all gradients of curves of this form behave identically to those of y=x^2, i.e. modulus of gradient increases with modulus of x, negative to the left of y-axis and positive to the right of the y-axis, am i correct?

#### Unknown008

Yes.

If you have learnt derivatives yet, you'll know that the derivative of a curve gives the equation of the gradient at any point. Here, the derivative of this curve is 2x. So, the gradient is directly proportional to the magnitude of x, which you got right. #### Sudharaka

make a generalisation about the gradient at any point on the graph of y=x^2 + c, where c is any real number

my textbook gives no answer for this, so i'd just like to make sure that the correct answer is that all gradients of curves of this form behave identically to those of y=x^2, i.e. modulus of gradient increases with modulus of x, negative to the left of y-axis and positive to the right of the y-axis, am i correct?
Dear furor celtica,

You could find the gradient using calculus.

$$\displaystyle y=x^2+C\Rightarrow{\frac{dy}{dx}=2x}$$

Therefore the gradient of $$\displaystyle y=x^2+C$$ at any point is identical to the gradient of $$\displaystyle y=x^2$$