Question

What is the range of the function `y= -3x² + 6x +4`?

Answer 1

Solution 1.

The y value of the turning point will determine the range of the equation.

Use the formula `x=-b/(2a)` to find the x value of the turning point.

Substitute in the values from the equation;

`x=(-(6)) / (2(-3))`

`x=1`

Substitute `x=1` into the original equation for the `y` value.

`y=-3(1)^2 + 6(1) + 4`

`y=7`

Since the `a` value of the quadratic is negative, the turning point of the parabola is a maximum. Meaning all `y` values less than 7 will fit the equation.

So the range is `y≤ 7`.

Solution 2.

You can find the range visually by graphing the parabola. The following graph is for the equation `-3x^2 + 6x + 4`

graph{-3x^2 + 6x + 4 [-16.92, 16.94, -8.47, 8.46]}

We can see that the maximum value of y is 7. Therefore, the range of the function is `y≤ 7`.