Given \( f(x)=x^{2} \), after performing the following transformations: shift upward 34 units and shift 35 units to the right, the new function \( g(x)=\square \)

To find the new function \( g(x) \) after the specified transformations on \( f(x) = x^2 \), we first handle the horizontal shift (35 units to the right). This transformation will replace \( x \) with \( x - 35 \), yielding \( (x - 35)^2 \). Next, we apply the vertical shift by adding 34 units upwards, which results in the final expression: \[ g(x) = (x - 35)^2 + 34 \] So, \( g(x) = (x - 35)^2 + 34 \). Now, let's spice things up with some interesting nuggets of information! Transformations in functions can make for pretty extravagant parties! Did you know that shifting a function alters its graph without changing its shape? It's like moving a painting on the wall, but the art itself stays the same. So, whether you’re sending your parabolas on an adventure to the right and up, they fundamentally retain their “parabolic identity.” If you're ever grappling with these transformations, remember: horizontal shifts involve shifting the input (x) while vertical shifts affect the output (the function’s value). A common trap is forgetting the sign change when shifting horizontally. Focus on that trusty old \(x - h\) form—you’ll cruise through transformations with ease!