y

Given the information about the x-intercepts $(-2,0)$ and $(1,0)$ and the y-intercept $(0,3)$, we can construct the quadratic equation in the factored form as follows:

The general form of a quadratic equation with known x-intercepts is:

$y = a(x - x_1)(x - x_2)$

where $x_1$ and $x_2$ are the x-intercepts.

For the x-intercepts $(-2,0)$ and $(1,0)$, the equation becomes:

$y = a(x + 2)(x - 1)$

Now, use the y-intercept $(0,3)$ to find the value of $a$. Substituting $x = 0$ and $y = 3$ into the equation:

$3 = a(0 + 2)(0 - 1)$

$3 = a(2)(-1)$

$3 = -2a$

Solving for $a$:

$a = -\frac{3}{2}$

Thus, the equation of the quadratic is:

$y = -\frac{3}{2}(x + 2)(x - 1)$

Expanding this equation:

$y = -\frac{3}{2}(x^2 + x - 2)$

$y = -\frac{3}{2}x^2 - \frac{3}{2}x + 3$

So, the quadratic equation is:

$y = -\frac{3}{2}x^2 - \frac{3}{2}x + 3$

Would you like further details or clarification on any steps?

Here are 5 related questions:

1. How do you factor a quadratic equation from a graph?
2. What is the significance of the y-intercept in a quadratic equation?
3. Can you solve the quadratic equation using the quadratic formula instead?
4. How do you determine the vertex of a quadratic equation from its graph?
5. What role does the coefficient $a$ play in the shape of the parabola?

Tip: The sign of $a$ in a quadratic equation determines if the parabola opens upward or downward!