Determine a quadratic function in vertex form given each set of characteristics.
* minimum value of -24 and x-intercepts at -21 and -5
I have:
(-21,0) and (-5,0)
How would I find the x-coordinate of the vertex?
(Thank you)
Because of symmetry, the vertex is midway between the roots: x = -13
To find the x-coordinate of the vertex, you can use the formula:
x = -b / (2a)
where a and b are the coefficients of the quadratic function in vertex form. To determine these coefficients, you can plug in the x-intercepts (-21 and -5) into the vertex form equation, which is:
f(x) = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex. Since the x-intercepts are at -21 and -5, we can plug in these values to find two equations:
0 = a(-21 - h)^2 + k
0 = a(-5 - h)^2 + k
From the first equation, we can simplify it by expanding the squares and simplifying further:
0 = a(h^2 + 42h + 441) + k
With the second equation, we can do the same:
0 = a(h^2 + 10h + 25) + k
Now, we can create a system of equations using these two equations:
a(h^2 + 42h + 441) + k = 0
a(h^2 + 10h + 25) + k = 0
Next, we can solve this system of equations simultaneously to find the values of a, h, and k. Once you have the values of h and k, you can substitute them into the formula x = -b / (2a) to find the x-coordinate of the vertex.