Best answer will be branliest and win 10 points! Please helpppp
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The function in vertex form is ⇒ 3rd answer
Step-by-step explanation:
The vertex form of the quadratic function f(x) = ax² + bx + c is
f(x) = a(x - h)² + k, where
- a is the coefficient of x²
- (h , k) are the coordinates of the vertex point
, wher b is the coefficient of x
- k = f(h), that means value f(x) when x = h
∵ f(x) = x² + x + 1
∴ a = 1 , b = 1
∵
- Substitute the values of a and b to find h
∴
∴
Substitute the value of x in f(x) by the value of h to find k
∵ f( ) =
∴ f( ) =
∴ f( ) =
- k is the value of f(x) when x = h
∵ h =
∴ k = f( )
∴ k =
Substitute the values of a, h and k in the vertex form
∵ f(x) = a(x - h)² + k
∵ a = 1 , ,
∴
∴
The function in vertex form is
Learn more:
You can learn more about the quadratic functions in brainly.com/question/9390381
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Answer:
the radius 1 < r < 3, so r = 1.39
Step-by-step explanation:
notice that you have 3 cm radius for the big circle
r radius for small
and you have this rectangle with dimension 6 cm by 8cm
so if you can imagine the radius segments moving into vertical and horizontal position
there are two boxes... 6cm by 6cm and 2r by 2r, but 6cm + 2r is not 8cm
diagonal of rectangle = root(8*8 + 6*6) = 10 cm
diagonals intersect at ((0 + 8)/ 2, (0 + 6) / 2)= (4, 3) but center of big circle at (3, 3)
It appears that the point (4, 3) is on the diagonal but does not go through the centers of the circles. but I do see
slope of diagonal is 3/4 so if rise is 2r, then the run.
(2r/x) = 3/4
(4/3)* 2r = x
equation of diagonal : y = (3/4)x + 0 if the bottom left is (0,0)
equation of circle r : (x - (8-r))^2 + (y - (6 - r))^2 = r*r
---------
Use a trig function to find the length of this line segment from the center of the big circle to the intersection of the small circle.
The triangle has lengths 1 cm , 3cm and a segment called x
x = line segment from the center of the big circle to the intersection of the small circle.
the angle between 1cm and x is 180 - arctan (3/4) = 180 - 36.8698 deg.
=143.1302 degrees
trig formula:
c^2 = a^2 + b^2 - 2ab*cos C
c^2 = 1^2 + 3^2 - 2*1*3*cos (81.8698 )
c^2 = 10 - 6 cos (81.8698)
c^2 = 9.15146173
c = 3.0251383 units
ok so we know the diagonal is 10
the distance from the intersection to the corner is 10 - 5 - 3.0251383
= 1.9748616 units
we can say 1.9748 = root(2) * r
so that r = 1.9748 / root(2)
