Determine a quadratic function with this set of characteristics.
x-intercepts of 2 and 7 and maximum value of 25
( How would I find the x-coordinate for the vertex?)
(Thank you!)
given the x-intercepts,
y = a(x-2)(x-7)
The vertex is midway between the roots, at x = 9/2
y(9/2) = a(5/2)(-5/2) = -25/4 a
Since we want y(9/2) = 25, a = -4
y = -4(x-2)(x-7)
See
http://www.wolframalpha.com/input/?i=-4%28x-2%29%28x-7%29
25 is the maximum value
I used x = 9/2 and I got this equation:
-4(x-4.5)^2+25
To find the x-coordinate of the vertex, you can use the formula for the axis of symmetry, which is given by x = -b/2a, where a and b are the coefficients of the quadratic function in the form of ax^2 + bx + c.
In this case, we know that the x-intercepts are 2 and 7. The x-intercepts occur when the quadratic function equals zero. Therefore, we can set up two equations using the given x-intercepts:
1) When x = 2, the quadratic function equals zero:
a(2)^2 + b(2) + c = 0
2) When x = 7, the quadratic function equals zero:
a(7)^2 + b(7) + c = 0
Next, since the maximum value of the quadratic function is 25, we know that the x-coordinate of the vertex occurs at the axis of symmetry. Let's call this x-coordinate "h". Therefore, we can substitute x = h into the equation to find the maximum value:
a(h)^2 + b(h) + c = 25
These three equations can be used to solve for the coefficients a, b, and c of the quadratic function.
By substituting the values 2 and 7 into the first two equations, you can set up a system of equations and solve them simultaneously to find the values of a and b. Once you have the values of a and b, you can substitute them along with the x-coordinate of the vertex (h) into the third equation to find the value of c.
Once you have the values of a, b, and c, you can write the quadratic function in the form of ax^2 + bx + c.