Answer:
[tex]c = 25[/tex]
Step-by-step explanation:
Vertex of a quadratic function:
Suppose we have a quadratic function in the following format:
[tex]f(x) = ax^{2} + bx + c[/tex]
It's vertex is the point [tex](x_{v}, y_{v})[/tex]
In which
[tex]x_{v} = -\frac{b}{2a}[/tex]
[tex]y_{v} = -\frac{\Delta}{4a}[/tex]
Where
[tex]\Delta = b^2-4ac[/tex]
If a>0, the vertex is a minumum point, that is, the minimum value happens at [tex]x_{v}[/tex], and it's value is [tex]y_{v}[/tex].
In this question:
[tex]a = 1[/tex]
Minimum value at [tex]x = 4[/tex] means that [tex]x_v = 4[/tex]. So
[tex]x_{v} = -\frac{b}{2a}[/tex]
[tex]4 = -\frac{b}{2}[/tex]
[tex]b = -8[/tex]
So
[tex]f(x) = x^2 - 8x + c[/tex]
Minimum value of 9:
[tex]y_v = 9[/tex]
So
[tex]y_{v} = -\frac{\Delta}{4a}[/tex]
[tex]9 = -\frac{\Delta}{4}[/tex]
[tex]-\Delta = 36[/tex]
[tex]\Delta = -36[/tex]
[tex]b^2 - 4ac = -36[/tex]
[tex](-8)^2 - 4c = -36[/tex]
[tex]-4c = -100[/tex]
[tex]4c = 100[/tex]
[tex]c = \frac{100}{4}[/tex]
[tex]c = 25[/tex]
