The equation shown in the graph is $y = -x^2 + 6x$. To find the average rate of change of $y$ between $x = 0$ and $x = 1$, we can use the formula:

$\text{Average rate of change} = \frac{y(x_2) - y(x_1)}{x_2 - x_1}$

where $x_1 = 0$ and $x_2 = 1$.

First, let's find the values of $y$ at $x = 0$ and $x = 1$:

1. For $x = 0$: $y(0) = -(0)^2 + 6(0) = 0$

2. For $x = 1$: $y(1) = -(1)^2 + 6(1) = -1 + 6 = 5$

Now, using the average rate of change formula:

$\text{Average rate of change} = \frac{y(1) - y(0)}{1 - 0} = \frac{5 - 0}{1 - 0} = 5$

Thus, the average rate of change of $y$ between $x = 0$ and $x = 1$ is $5$.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the instantaneous rate of change of $y$ at $x = 1$?
2. How would you find the vertex of the parabola $y = -x^2 + 6x$?
3. What is the significance of the sign of the rate of change?
4. How do you determine the concavity of the graph from the equation?
5. Can you find the x-intercepts of the parabola?

Tip: The average rate of change of a function over an interval is the slope of the secant line that connects the points on the graph corresponding to the endpoints of the interval.