Step 1 :To find the y-intercept, plug in x=0: \(f(0) = -4(0)^2 - 40(0) = 0\). So, the point (0, 0) is where the graph of the function crosses the y-axis.
Step 2 :To find the x-intercepts, plug in f(x) = 0: \(0=-4 x^{2}-40 x\). Factor the equation: \(0=(-4x)(x+10)\). So, the function crosses the x-axis at (0,0) and (-10,0).
Step 3 :The parabola's axis of symmetry has the equation \(x=\frac{-b}{2a}\), where a = -4 and b = -40, so \(x=\frac{-(-40)}{2(-4)}=-5\). Thus, the axis of symmetry has the equation \(x=-5\).
Step 4 :To find the maximum point, find \(f(-5)=-4(-5)^2 - 40(-5)=(-4)(25)+200=-100+200=100\). So, the maximum point of the parabola is at \((-5, 100)\).
Step 5 :The parabola opens upward if a is positive. In this case, a = -4, meaning the parabola opens downward.