To find the equation of the parabola that passes through the points $(0, 4)$, $(1, 0)$, and $(-2, 0)$, we will assume that the equation of the parabola is of the form:

$y = ax^2 + bx + c$

Where $a$, $b$, and $c$ are constants we need to determine. Given that the parabola passes through the points, we can substitute the coordinates of the points into this equation and create a system of linear equations to solve for $a$, $b$, and $c$.

Step 1: Use the point $(0, 4)$

Substitute $x = 0$ and $y = 4$ into the general form of the parabola:

$4 = a(0)^2 + b(0) + c \quad \Rightarrow \quad c = 4$

So, the equation simplifies to:

$y = ax^2 + bx + 4$

Step 2: Use the point $(1, 0)$

Substitute $x = 1$ and $y = 0$ into the equation:

$0 = a(1)^2 + b(1) + 4 \quad \Rightarrow \quad a + b + 4 = 0$ $a + b = -4 \quad \text{(Equation 1)}$

Step 3: Use the point $(-2, 0)$

Substitute $x = -2$ and $y = 0$ into the equation:

$0 = a(-2)^2 + b(-2) + 4 \quad \Rightarrow \quad 4a - 2b + 4 = 0$ $4a - 2b = -4 \quad \text{(Equation 2)}$

Step 4: Solve the system of equations

Now, we have the following system of equations:

1. $a + b = -4$
2. $4a - 2b = -4$

Solve Equation 1 for $b$:

$b = -4 - a$

Substitute into Equation 2:

$4a - 2(-4 - a) = -4$ $4a + 8 + 2a = -4$ $6a + 8 = -4$ $6a = -12$ $a = -2$

Substitute $a = -2$ into $b = -4 - a$:

$b = -4 - (-2) = -4 + 2 = -2$

Step 5: Write the final equation

Now that we have $a = -2$, $b = -2$, and $c = 4$, the equation of the parabola is:

$y = -2x^2 - 2x + 4$

Final Answer:

The equation of the parabola is:

$y = -2x^2 - 2x + 4$

Would you like more details on how to solve this, or do you have other questions?

Further Questions:

1. What is the vertex of the parabola $y = -2x^2 - 2x + 4$?
2. How can we find the axis of symmetry for this parabola?
3. How do you determine the direction (upward or downward) of the parabola from the equation?
4. How do you find the focus and directrix of this parabola?
5. How can you use the quadratic formula to find the roots of the parabola equation?

Tip:

To find the vertex of a parabola given its equation $y = ax^2 + bx + c$, use the formula for the $x$-coordinate of the vertex: $x = -\frac{b}{2a}$.