$$ y = - \frac { 1 } { 39 } x ^ { 2 } + 30 \{ - 34 \leq x \leq 34 \} $$

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The maximum value of the quadratic function is 30, occurring at the vertex (0, 30). The given equation is a quadratic equation in the form $$ y = ax^2 + bx + c $$, where $$ a = -\frac{1}{39} $$, $$ b = 0 $$, and $$ c = 30 $$. The domain of the equation is given as $$ -34 \leq x \leq 34 $$. To find the maximum or minimum value of the quadratic function, we need to determine the vertex of the parabola represented by the equation. The vertex of a parabola in the form $$ y = ax^2 + bx + c $$ is given by the formula $$ x = -\frac{b}{2a} $$. In this case, since $$ b = 0 $$, the vertex is at $$ x = 0 $$. Substitute $$ x = 0 $$ into the equation to find the corresponding y-coordinate: $$ y = -\frac{1}{39}(0)^2 + 30 = 30 $$ Therefore, the maximum value of the quadratic function is 30, which occurs at the vertex (0, 30).

\( y = - \frac { 1 } { 39 } x ^ { 2 } + 30 \{ - 34 \leq x \leq 34 \} \)

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\( y = - \frac { 1 } { 39 } x ^ { 2 } + 30 \{ - 34 \leq x \leq 34 \} \)

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