The wife and I were doing homework together, and we noticed something really strange when charting quadratics with a TI-series graphing calculator:
f(5) = -x^2 + 110x - 1000
f(5) = -5^2 + (110*5) - 1000
f(5) = -25 + 550 - 1000
f(5) = -475
// Wait a minute...
-5^2 = -25 // Negative?
We knew this wasn't right, so we tried the formula out on an online calculator, and we got the same result:
So we decided to wrap the coefficient in parentheses, and it worked as expected:
// Wrap in parentheses...
(-5)^2 = 25 // Positive, as expected
Obviously, I think the second solutions must be correct... but I can't imagine that in today's day and age, I have to explicitly wrap every negative coefficient in parentheses to ensure proper evaluation on a calculator. Is this the case, or is the first evaluation actually correct?
Thanks for taking the time!
Modern calculators follow the appropriate precedence of operations: exponentiation goes before products, products go before additions. If you type "2+3*5", my calculator (TI-83+) correctly gives 17 as the answer. When you type "-5^2", the calculator correctly performs the square first, then multiplies by $-1$.
Note that if you simply write $-5^2$, then this does mean $-(5^2)$, and not $(-5)^2$, because of the precedence of the operations. When you write $-x^2$, you mean $-(x^2)$, not $(-x)^2$.
The function $f(x) = -x^2 + 110 x - 1000$ is the function $$f(x) = -\left( x^2\right) + \left( 110 x\right) - 1000,$$ and as such, its value at $5$ is $$-(5^2) + (110\times 5) - 1000 = -25 + 550 - 1000 = -475.$$
If the function you meant to write was $$g(x) = (-x)^2 + 110x - 1000 = x^2 + 110x - 1000,$$ then you should have written that.
The calculator correctly evaluated what was typed; whether what was typed was what was meant is of course a separate matter.