1. The problem states that the frog's jump is modeled by the function $f(x) = -x^2 + 2x$, and we need to find the maximum height of the jump. 2. This function is a quadratic in the form $f(x) = ax^2 + bx + c$ where $a = -1$, $b = 2$, and $c = 0$. 3. Since $a < 0$, the parabola opens downward, so the vertex represents the maximum point. 4. The $x$-coordinate of the vertex is given by the formula $$x = -\frac{b}{2a} = -\frac{2}{2 \times (-1)} = 1.$$ 5. Substitute $x = 1$ back into the function to find the maximum height: $$f(1) = -(1)^2 + 2(1) = -1 + 2 = 1.$$ 6. Therefore, the maximum height of the frog's jump is $1$ unit. Final answer: 1