Hello! Please explain how to solve this step-by-step. (Question 111.) I tried solving but not sure if I did it correctly. Thanks!

Step-by-step explanation:

We want to find the equation of a parabola of the form:

This passes through (0,1) and is tangent to the line:

At the point (1,0).

First, note that (0,1) is the y-intercept. In other words, our constant c is 1. Thus:

From the equation of the tangent line, we can see that it has a slope of one.

Recall that the slope of the tangent line at a point is equivalent to the value of the derivative at the same point.

In other words, the value of the derivative of our parabola at x = 1 must be one.

Find the derivative. Take the derivative of both sides with respect to x:

Expand:

Use the Power Rule. Since we're differentiating with respect to x, we can treat a and b as constants. Thus:

Now, since the slope of the tangent line at x = 1 is 1, this means that:

Simplify:

Let's hold on to this equation for now.

Since the line is tangent at the point (1,0), this means that our original function equals zero when x = 1. In other words:

Simplify:

This yields the following system of equations:

Solve for a and b.

From our previous equation, let's subtract 2a from both sides:

Substitute this into the newly acquired equation:

Solve for a. Rewrite:

Combine like terms:

Hence:

Find b:

Therefore, a = 2 and b = -3.

Then by substitution, we can see that our final equation is:

step-by-step explanation: