From the graph and the equation $y = 3x - x^2$, let's determine the intercepts.

Finding the x-intercepts:

The x-intercepts occur where $y = 0$. Setting the equation to zero: $0 = 3x - x^2$ $x^2 - 3x = 0$ Factoring out $x$: $x(x - 3) = 0$ Setting each factor equal to zero: $x = 0 \quad \text{and} \quad x = 3$

Finding the y-intercept:

The y-intercept occurs where $x = 0$. Substituting into the equation: $y = 3(0) - 0^2 = 0$

Solutions:

• x-intercept (smaller x-value): $(0, 0)$
• x-intercept (larger x-value): $(3, 0)$
• y-intercept: $(0, 0)$

These answers are consistent with the graph provided. If you have any questions or need further details, feel free to ask.

Related Questions:

1. How do you determine the vertex of the parabola $y = 3x - x^2$?
2. What is the axis of symmetry for the equation $y = 3x - x^2$?
3. How can you verify the intercepts algebraically without a graph?
4. What transformations have been applied if the original equation was $y = -x^2$?
5. How would the graph change if the equation was $y = x^2 - 3x$?

Tip:

When you have a quadratic equation in the form $y = ax^2 + bx + c$, setting $y = 0$ and factoring can directly give you the x-intercepts.