Answer: To find the maximum value of the y-intercept of a quadratic function, you need to determine the equation of the quadratic function in vertex form. The vertex form of a quadratic function is given by:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
In this case, you're given the x-intercepts at (1, 0) and (5, 0). This means that the roots of the quadratic equation are x = 1 and x = 5. You can write the equation as:
f(x) = a(x - 1)(x - 5)
Now, you also know that the y-intercept is at (0, -5), so you can plug these coordinates into the equation:
-5 = a(0 - 1)(0 - 5)
Simplify this equation:
-5 = -5a
Now, solve for "a" by dividing both sides by -5:
a = 1
So, the equation of the quadratic function is:
f(x) = (x - 1)(x - 5)
To find the y-intercept (the maximum value of the function), you need to find the vertex of this parabola. The vertex of a quadratic in the form f(x) = a(x - h)^2 + k is at (h, k).
In this case:
h = (1 + 5) / 2 = 6 / 2 = 3
k = f(3) = (3 - 1)(3 - 5) = 2(-2) = -4
So, the vertex is at (3, -4), and the maximum value (the y-intercept) is -4. Therefore, the maximum value of the y-intercept is -4.