Question 1167615
your problem is shown below:


<img src = "http://theo.x10hosting.com/2020/101809.jpg" >
let y = f(x).
the function becomes y = -x^2 + 8x - 7


the standard form of a quadratic equation is y = ax^2 + bx + c
a is the coefficient of the x^2 term.
b is the coefficient of the x term.
c is the constant term.


the standard form of the equation becomes y = -x^2 + 8x - 7


the vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
a is the coefficient of the x^2 term,
(h,k) is the coordinates of the vertex.


from the standard form of the equation, you would solve for the x-coordinate of the vertex by the following equation:
x = -b/(2a)
when b = 8 and a = -1, this becomes x = -8/(-2) = 4.
the y-coordinate of the vertex is found by replacing x in the standard form of the equation by 4 and solving for y.
you get y = -4^2 + 8*4 - 7 which becomes y = -16 + 32 - 7.
solve for y to get y = 9.


the vertex from the standard form of the equation is (x,y) = (4,9).
the vertex from the vertex form of the equation is (h,k) = (4,9).


the vertex form of the equation becomes y = -1 * (x-4)^2 + 9.


both standard form of the equation and vertex form of the equation can be seen in the following graph.



<img src = "http://theo.x10hosting.com/2020/101807.jpg" >


since both equations are equivalent, they both draw the same graph.


when you bring the equations down 5 units, you just subtract 5 from the constant term in the standard equation and you drop 5 from the value of k in the vertex form of the equation.


you get:


y = -x^2 + 8x - 12 in the standard form.
y = -(x-4)^2 + 4 in the vertex form.


both standard form of the equation and vertex form of the equation can be seen in the following graph.


<img src = "http://theo.x10hosting.com/2020/101808.jpg" >


you can see from both graphs that the vertex was dropped 5 units from (4,9) to (4,4).


here's a reference you might find helpful.


<a href = "https://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php" target = "_blank">https://www.mathwarehouse.com/geometry/parabola/standard-and-vertex-form.php</a>