The graph of the function y = 4x² is a parabola that opens upwards. When we focus on the interval from 0 to 3, several key characteristics emerge that define its shape and behavior.
Y-Intercept
First, at x = 0, the value of y is 0. Therefore, the graph begins from the point (0, 0). This is the vertex of the parabola, which is its lowest point since it opens upwards.
Behavior at the Endpoint
Next, as we move to the right along the x-axis, we find where the graph is at x = 3. Here, the calculation shows:
y = 4(3)² = 4(9) = 36This means that at x = 3, the graph reaches the point (3, 36).
Shape of the Graph
The shape of the graph between these points is a smooth curve that increases continuously. As x increases from 0 to 3, the value of y rises steeply. The rate of increase is quadratic, which means the graph will get significantly steeper as it moves further right.
Conclusion
In summary, the graph of y = 4x² on the interval from 0 to 3 starts at the origin (0, 0) and ends at the point (3, 36), forming a typical upward-opening parabola that steepens as it moves along the x-axis. This visual representation effectively illustrates the quadratic nature of the equation, highlighting both the gradual and rapid increases in y values as x progresses.