Question

IncorrectYour answer Is Incorrect.Find the range of the quadratic function.y=3x2 - 30x + 77Write your answer as an inequality using x or y as appropriate.Or, you may instead click on "Empty set" or "All reals" as the answer.DSODO음EmptysetAll realsХ?

Answer 1

We are given the following quadratic function:

`y=3x^2-30x+77`

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:

`y=ax^2+bx+c`

The x-coordinate of the vertex is given by:

`x=-(b)/(2a)`

Replacing we get:

`x=-(-30)/(2(3))`

Solving we get:

`x=(10)/(2)=5`

The y-coordinate of the vertex is found by replacing this value in the quadratic equation:

`\begin{gathered} y=3(5)^2-30(5)+77 \\ y=3(25)-150+77 \\ y=2 \end{gathered}`

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:

`R=\mleft\lbrace y\in\R\parallel y\ge2\rbrace\mright?`