### Video Transcript

Determine the average rate of change function π΄ of β for π of π₯ is equal to six π₯ squared minus three at π₯ is equal to one.

The average rate of change function π΄ of β for a function π of π₯ at π₯ is equal to π₯ one is given by π΄ of β is equal to π at π₯ is equal to π₯ one plus β minus π at π₯ is equal to π₯ one over β, where β is a small change in π₯. Weβve been given the function π of π₯ is six π₯ squared minus three and asked to determine the rate of change function π΄ of β at π₯ is equal to one. This means that in our average rate of change function π΄ of β, π₯ one is equal to one. So, we have the function π of π₯ is six π₯ squared minus three and π₯ one is equal to one. And our average rate of change function π΄ of β is equal to π at π₯ is equal to one plus β minus π at π₯ is equal to one all over β.

Of course, we havenβt finished yet. We need to evaluate our function π at π₯ is equal to one plus β and π at π₯ is equal to one. So, letβs do this. Replacing π₯ with one plus β in our function π gives us six times one plus β squared minus three. That is six times one plus two β plus β squared all minus three, which is six plus 12β plus six β squared minus three. And collecting like terms and rearranging gives us six β squared plus 12β plus three.

Now, we evaluate π at π₯ is equal to one. We have six times one squared minus three, which is six minus three, and thatβs equal to three. We have π at π₯ is equal to one plus β is six β squared plus 12β plus three and π at π₯ is equal to one is equal to three, which we can now substitute into our average rate of change function π΄ of β. This gives us π΄ of β is equal to six β squared plus 12β plus three minus three all over β.

The positive three cancels with the negative three so that we have six β squared plus 12β over β. We have a common factor of β in the numerator. If we rewrite this as β times six β plus 12 over β, we can cancel this β in the numerator with the β in the denominator, and weβre left with six β plus 12.

The average rate of change function π΄ of β for π of π₯ is equal to six π₯ squared minus three at π₯ is equal to one is six β plus 12.