Questions: Find the vertex of the given parabola. Give all values correct to 3 decimal places. y=9x^2-9x+3 ( , )

Transcript text: Find the vertex of the given parabola. Give all values correct to 3 decimal places. \[ y=9 x^{2}-9 x+3 \] ( $\square$ , $\square$ ) Submit Answer

Solution

Solution Steps

To find the vertex of a parabola given by the equation $ y = ax^2 + bx + c $, we use the vertex formula $ x = -\frac{b}{2a} $. Once we have the x-coordinate, we substitute it back into the equation to find the y-coordinate.

Step 1: Identify the coefficients

Given the quadratic equation: \[ y = 9x^2 - 9x + 3 \] we identify the coefficients: \[ a = 9, \quad b = -9, \quad c = 3 \]

Step 2: Calculate the x-coordinate of the vertex

The x-coordinate of the vertex of a parabola $ y = ax^2 + bx + c $ is given by: \[ x = -\frac{b}{2a} \] Substituting the values of $ a $ and $ b $: \[ x = -\frac{-9}{2 \cdot 9} = \frac{9}{18} = 0.5 \]

Step 3: Calculate the y-coordinate of the vertex

To find the y-coordinate, substitute $ x = 0.5 $ back into the equation: \[ y = 9(0.5)^2 - 9(0.5) + 3 \] \[ y = 9 \cdot 0.25 - 4.5 + 3 \] \[ y = 2.25 - 4.5 + 3 = 0.75 \]