Respuesta :
We are given the following quadratic function:
[tex]y=3x^2-30x+77[/tex]To determine the range we need first to determine the vertex of the quadratic function. To do that, since we have an equation of the form:
[tex]y=ax^2+bx+c[/tex]The x-coordinate of the vertex is given by:
[tex]x=-\frac{b}{2a}[/tex]Replacing we get:
[tex]x=-\frac{-30}{2(3)}[/tex]Solving we get:
[tex]x=\frac{10}{2}=5[/tex]The y-coordinate of the vertex is found by replacing this value in the quadratic equation:
[tex]\begin{gathered} y=3(5)^2-30(5)+77 \\ y=3(25)-150+77 \\ y=2 \end{gathered}[/tex]Now, since the term "a" is a positive number the parabola opens upwards, and the range is the values of "y" that are larger than the y-coordinate of the vertex, that is:
[tex]R=\mleft\lbrace y\in\R\parallel y\ge2\rbrace\mright?[/tex]