Substituting $2$ for $x$ yields an expression that represents all possible heights of the second arrow when it reaches the castle wall: Setting up and solving an inequality will reveal all such values of $k$.įirst, identify $f(x + k)$ as defined in this particular case: Part b reveals one value ($-2$) of $k$ for which the function $f(x + k)$ would result in the arrow clearing the wall. Since $f$ is defined as $f(x) = 6 - x^2$, the function $f(x - 2)$ is defined as follows: A function that subtracts $2$ from every input value before following the procedures of $f$ would accomplish this-namely, $f(x - 2)$.
The user must enter a function whose output is $f(0)$ when the input is $x=2$. This is accomplished by moving the entire graph two units to the right, essentially moving the archer two units closer to the castle wall. The maximum value must occur at $x=2$ in order for the arrow to clear the castle wall by the greatest margin. The maximum value of $f(x)$ is $6$, which occurs at $x=0$.
