Question: The graph represents a quadratic function. Write an equation of the function in standard form.
$(-4, 0)$
$(-0.5, 0)$
$(0, 4)$
y = ax^2 + bx + c
1. **State the problem:** We need to find the quadratic function $y = ax^2 + bx + c$ that passes through the points $(-4, 0)$, $(-0.5, 0)$, and $(0, 4)$.
2. **Use the fact that the points $(-4, 0)$ and $(-0.5, 0)$ are roots:** Since these are x-intercepts, the quadratic can be written in factored form as:
$$y = a(x + 4)(x + 0.5)$$
3. **Use the point $(0, 4)$ to find $a$:** Substitute $x=0$ and $y=4$ into the equation:
$$4 = a(0 + 4)(0 + 0.5) = a \times 4 \times 0.5 = 2a$$
4. **Solve for $a$:**
$$4 = 2a \implies a = \frac{4}{2} = 2$$
5. **Write the equation with $a=2$:**
$$y = 2(x + 4)(x + 0.5)$$
6. **Expand to standard form:**
$$y = 2(x^2 + 0.5x + 4x + 2) = 2(x^2 + 4.5x + 2)$$
$$y = 2x^2 + 9x + 4$$
7. **Final answer:**
$$\boxed{y = 2x^2 + 9x + 4}$$
This is the quadratic function in standard form that fits the given points.