Answer
See explanation
Work Step by Step
Using a graphing utility, the graph is as shown.
The vertex is $(3,-5)$.
The axis of symmetry is $x=3$.
There is no $x$-intercept.
Finding the standard form of the quadratic function:
$$f(x)=-4x^2+24x-41$$ $$f(x)=-4(x^2-6x)-41$$ $$f(x)=-4\left(x^2-6x+\left(\frac{-6}{2}\right)^2\right)-41+4\left(\frac{-6}{2}\right)^2$$ $$f(x)=-4(x^2-6x+9)-41+36$$ $$f(x)=-4(x-3)^2-5$$
Thus, from the form $y=a(x-h)^2+k$, the vertex is $(3,-5)$ and axis of symmetry is $x=3$.
Finding the $x$-intercept by setting $y=0$:
$$0=-4(x-3)^2-5$$ $$5=-4(x-3)^2$$ $$-4(x-3)^2=5$$ $$(x-3)^2=-\frac{5}{4}$$ $$x-3=\sqrt{-\frac{5}{4}}~~imaginary$$
Thus, there is no $x$-intercept.
