Consider the function 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏. If 𝑓 of zero equals four and 𝑓 of one equals seven, find the value of 𝑓 of negative two.
We know that our function is in the form 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏. That means that 𝑓 of zero would be 𝑎 times zero squared plus 𝑏. And 𝑓 of zero equals four. Zero squared equals zero. And zero times 𝑎 would be equal to zero. And that means zero plus 𝑏 equals four. 𝑏 equals four. We also know 𝑓 of one which would be 𝑓 of one equals 𝑎 times one squared plus 𝑏. If 𝑓 of one equals seven, one squared equals one. One times 𝑎 equals 𝑎. Seven equals 𝑎 plus 𝑏. But we’ve already found that 𝑏 equals four. So we plug that in.
To solve for 𝑎, we can subtract four from both sides. Seven minus four is three. Three equals 𝑎. Or the way we more commonly write it, 𝑎 equals three and 𝑏 equals four. So we plug in what we know to our original equation 𝑓 of 𝑥 equals three 𝑥 squared plus four. And now we solve for 𝑓 of negative two which will be three times negative two squared plus four. Negative two squared equals four. Three times four plus four becomes 12 plus four which is 16. 𝑓 of negative two equals 16.