A curve is such that = 2(kx-1)^5 where k is a constant. Given that the curve passes through points ( 0 , 1) , ( 1 , 8 ) find the equation of the curve.

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Explanation:

Multiply dx on each side to move it over.

Doing so gets us to

Next, integrate both sides with respect to the dy and dx terms

We can do a u-substitution here.

Let u = kx-1 which leads to du/dx = k which turns into and rearranges to

So the steps continuing forward could look like

Let's plug in (x,y) = (0,1) and do a bit of algebra and solve for C. We'll get some equation in terms of k on the right hand side

Now let's plug in (x,y) = (1,8) along with that C value. Then solve for k to get an actual number.

I'm skipping a bunch of steps, but you should get to when solving that equation. You can use a graphing calculator to find the root of that function (think of k as the input x) and the x intercept is 3.

This leads to,

Therefore,

If k = 3, then,

If you were to differentiate the answer function with respect to x, then you should get the dy/dx expression mentioned.

shift 4 units up, and 1 unit left.

it will translate f(x) = 3^x to g(x) = 3^(x+1) + 4

the answer is ! ?