Step-by-step explanation:
1.
the y-intercept comes automatically from the given form, as the y-intercept is the y value when x = 0.
so, in
x² - 12x + 46
the y-intercept is 46, or as point (0, 46).
now to the vertex form and vertex itself.
the vertex form for such a quadratic equation is
y = a(x - h)² + k
with (h, k) being the vertex.
since we have a plain x² term in the basic expression, "a" must be 1.
so, we have the simpler version
y = (x - h)² + k = x² - 2hx + h² + k
so, let's compare this with our basic expression, and we see
-2h = -12
h = 6
h² + k = 46
6² + k = 46
36 + k = 46
k = 10
so, the vertex form is
y = (x - 6)² + 10
and the vertex is (6, 10)
2.
the equating of a circle is
(x - x1)² + (y - y1)² = r²
with (x1, y1) being the center of the circle, and r being the radius.
so, the center is then (3, -3), and the radius is sqrt(36)=6.
3.
as explained in 2. this would be
(x - 4)² + (y + 9)² = 3² = 9
4.
x² + y² + 6x - 2y + 9 = 0
this has to turn into
(x - x1)² + (y - y1)² = r²
x² - 2x×x1 + x1² + y² - 2y×y1 + y1² = r²
x² - 2x×x1 + x1² + y² - 2y×y1 + y1² - r² = 0
"x² + 6x" suggests that for x1 of the first term
6x = - 2x×x1
x1 = -3
and "y² - 2y" suggests for y1 of the second term
-2y = -2y×y1
y1 = 1
that gives us for the constant terms and radius
x1² + y1² - r² = 9
(-3)² + 1² - r² = 9
9 + 1 - r² = 9
1 - r² = 0
r² = 1
r = 1
so, the standard circle equation is
(x + 3)² + (y - 1)² = 1
