Question Find the equation of the axis of symmetry of the following parabola algebraically, \[ y=-3 x^{2}+18 x-45 \] Answer

To find the equation of the axis of symmetry of the parabola given by the equation \(y = -3x^2 + 18x - 45\), we can use the formula for the axis of symmetry, which is given by \(x = -\frac{b}{2a}\). In this case, \(a = -3\) and \(b = 18\). Plugging in the values, we get: \[ x = -\frac{18}{2 \times -3} = -\frac{18}{-6} = 3. \] Thus, the equation of the axis of symmetry is \(x = 3\). The axis of symmetry is essentially a vertical line that splits the parabola into two mirror-image halves. This means that any point on one side of the parabola has a corresponding point on the other side at the same distance from the axis, making it a crucial concept when analyzing parabolas and their transformations! Another fun fact about parabolas is that they are not just mathematical constructs! They show up in real life in the design of satellite dishes and suspension bridge cables. The shape helps them achieve the most efficient form for focusing signals or distributing weight, showcasing the beauty of mathematics in the world around us!