A parabola has been drawn on a graph but there is information missing. The equation has not been written on the curve. It has a minimum turning point, a y intercept of +6 and cuts the x-axis at +2 and +#

Determine the equation to the curve

axis of symmetry is halfway between 2 and 3 or x =2.5

to get the zeros of y at x = 2 and 3
y = k (x-2)(x-3)
to get y = 6 at x = 0
6 = k (-2)(-3)
k = 1
so
y = x^2 -5 x + 6 (answer)

check if vertex is at x = 2.5 by completing the square
y -6 = x^2 - 5 x
y - 6 + (5/2)^2 = x^2 - 5 x + (5/2)^2
y + .25 = (x-2.5)^2
so vertex at (2.5 , -.25) checks :)

x-axis at +2 and +3

To determine the equation of the parabola, we can start by finding the coordinates of the turning point.

Since the parabola has a minimum turning point, the equation will be in the form of y = a(x - h)^2 + k, where (h,k) represents the coordinates of the turning point.

Given that the parabola cuts the x-axis at +2 and +# (which implies that the roots are 2 and #), we can use the fact that the roots of a quadratic equation can be found using the equation (x - r)(x - s) = 0, where r and s are the roots.

So, the equation for the parabola can be written as:

y = a(x - 2)(x - #)

Now let's find the coordinate of the turning point.

Since the y-intercept is +6, we know that when x = 0, y = 6. Substituting these values into the equation, we get:

6 = a(0 - 2)(0 - #)
6 = a(-2)(-#)
6 = 2a#

Also, the turning point lies on the axis of symmetry, which is the average of the two roots. So, the x-coordinate of the turning point is:

(h) = (2 + #)/2 = 1 + #/2

Therefore, the coordinate of the turning point is (1 + #/2, 6).

We can substitute these values into the equation y = a(x - h)^2 + k to solve for a.

6 = a[(x - (1 + #/2))^2 + 6]

Expanding and rearranging the equation, we get:

6 = a[(x - 1 - #/2)^2 + 6]
6 = a(x^2 - 2x(1 + #/2) + (1 + #/2)^2)

Now, we can expand further:

6 = a(x^2 - 2x + 2 - x# + # + #^2/4 + 6)
6 = a(x^2 - 2x - x# + # + #^2/4 + 8)

Combining like terms, we have:

6 = a(x^2 + (-2 - #)x + (#^2/4 + # + 8))

By comparing the coefficients on both sides, we can conclude that:

a = 1, -2 - # = 0, and #^2/4 + # + 8 = 0.

From the second equation, we can solve for #:

-2 - # = 0
# = -2

Substituting this value back into the third equation, we have:

(-2)^2/4 - 2 + 8 = 0
1 - 2 + 8 = 0
7 = 0.

Since the third equation gives us a false statement, there is no value for # that satisfies the equation.

Therefore, there is no equation that satisfies the given conditions.

To determine the equation of the curve, we need to use the given information and some fundamental concepts related to parabolas.

1. Minimum Turning Point: A parabola with a minimum turning point is in the form of "y = ax^2 + bx + c", where "a" is a positive value.
2. Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. Since the y-intercept is +6, we have the point (0, 6) on the parabola.
3. X-intercepts: The parabola cuts the x-axis at +2 and +# (unknown value). We have two x-intercepts, y = 0 when x = +2 and x = +#.
4. Vertex Form of the Parabola: The equation in vertex form for a parabola with vertex (h, k) is "y = a(x-h)^2 + k".

From the given information, we can deduce the following:

1. The y-intercept (0, 6) implies that c = 6.
2. The x-intercepts at x = +2 and x = +# suggest that the factors of the quadratic equation are (x - 2) and (x - #).

To find the value of "#", we can use the fact that the vertex of a parabola lies in the middle of its x-intercepts. In this case, the given minimum turning point should be between +2 and "#". Let's assume the value of "#" is "+4". If this assumption doesn't yield a consistent result, we can try with other reasonable values.

Now, using the vertex form of a parabola, we can determine the equation by following these steps:

Step 1: Find the x-coordinate of the vertex.
- The x-coordinate of the vertex is the average of the x-intercepts: (2 + 4) / 2 = 3.

Step 2: Substitute the vertex coordinates into the equation to solve for "a".
- (3, k) represents the vertex coordinates, and we know k = c = 6.
- Substitute the values into the equation: 6 = a(3 - 2)^2 + 6.
- Simplify: 6 = a + 6.
- Subtract 6 from both sides: a = 0.

Step 3: Write the equation.
- Now that we have obtained the value of "a" as 0, we can write the equation in vertex form:
- y = a(x -h)^2 + k.
- Plugging in the values: y = 0(x - 3)^2 + 6.
- Simplify: y = 6.

Therefore, the equation of the parabola is y = 6. Since the parabola has a minimum turning point and is parallel to the x-axis, the graph is a horizontal line passing through (0, 6).