Final Answer:
The graph of the function y = -34x² + 1542x - 10037 is a parabola that opens downwards, reaching its maximum point at x ≈ 1 and y ≈ -8495.75.
Step-by-step explanation:
The equation y = -34x² + 1542x - 10037 is in the form of a quadratic function, which means its graph is a parabola. The coefficient of the x² term (-34) is negative, indicating that the parabola opens downwards.
To analyze the graph further, we can find the vertex, which is the point where the parabola changes direction. We can do this by using the formula for the x-coordinate of the vertex:
x_vertex = -b / 2a
where a and b are the coefficients of the x² and x terms, respectively. In this case, a = -34 and b = 1542, so:
x_vertex = -1542 / (2 * -34) ≈ 1
To find the y-coordinate of the vertex, we can substitute this value of x back into the original equation:
y_vertex = -34(1)² + 1542(1) - 10037 ≈ -8495.75
Therefore, the vertex of the parabola is approximately at (1, -8495.75).