Respuesta :
To find the vertex (h,k), we have to find h using the following formula
[tex]h=-\frac{b}{2a}[/tex]Where a = 1 and b = -10.
[tex]h=-\frac{-10}{2\cdot1}=5[/tex]Then, we find k by evaluating the function when x = 5.
[tex]y=5^2-10\cdot5+9=25-50+9=-16[/tex]Hence, the vertex is (5,-16).
The axis of symmetry is given by the h coordinate of the vertex.
Hence, the axis of symmetry is x = 5.
The y-intercept is found when x = 0.
[tex]y=0^2-10\cdot0+9=9[/tex]The y-intercept is (0,9).
The x-intercepts are found when y = 0.
[tex]x^2-10x+9=0[/tex]To solve this expression, we have to look for two numbers which product is 9, and which addition is 10. Those numbers are 9 and 1.
[tex](x-9)(x-1)=0[/tex]Then, we use the zero product property to express both solutions
[tex]\begin{gathered} x-9=0\rightarrow x=9 \\ x-1=0\rightarrow x=1 \end{gathered}[/tex]Hence, the x-intercepts are (9,0) and (1,0).
The minimum value is defined by the k coordinate of the vertex.
Therefore, the minimum value of the function is -16.
The domain of the function would be all real numbers because quadratic functions don't have any domain restrictions.
[tex]D=(-\infty,\infty)[/tex]The range of the function is determined by the vertex, given that the parabola opens upwards, then the range is
[tex]R\colon\lbrack-16,\infty\rbrack[/tex]