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

Aug 8, 2017

Solution 1.

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

Use the formula $x = - \frac{b}{2 a}$ to find the x value of the turning point.

Substitute in the values from the equation;

$x = \frac{- \left(6\right)}{2 \left(- 3\right)}$

$x = 1$

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

$y = - 3 {\left(1\right)}^{2} + 6 \left(1\right) + 4$

$y = 7$

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 $- 3 {x}^{2} + 6 x + 4$

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.