Find the vertex of the graph of the quadratic function. \( f(x)=2 x^{2}-8 x+10 \)

To find the vertex of the quadratic function \( f(x) = 2x^2 - 8x + 10 \), we can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 2 \) and \( b = -8 \). Plugging in the values, we have: \[ x = -\frac{-8}{2(2)} = \frac{8}{4} = 2. \] Now, substitute \( x = 2 \) back into the function to find the corresponding \( y \)-value: \[ f(2) = 2(2)^2 - 8(2) + 10 = 8 - 16 + 10 = 2. \] Thus, the vertex of the graph is at the point \( (2, 2) \). Now, let's have a little fun with this! The vertex of a quadratic function is akin to the tip-top of a roller coaster—it's where the fun starts or ends! Here, at \( (2, 2) \), you can either be soaring up or gliding down, depending on whether you're going up or down the parabola. Just remember, this vertex holds the key to the function's minimum value since the parabola opens upwards (thanks to that positive coefficient in front of \( x^2 \))! Finding the vertex isn't just a math exercise—it's essential to understanding how to graph a quadratic equation accurately. A common mistake is miscalculating the vertex's x-value, often due to forgetting the negative in \( -\frac{b}{2a} \). Remember to carefully apply the formula and check your math. Happy graphing!