Question: 1 — Introduction to Quadratic Equations
LEARNING OBJECTIVE: Identify a parabola given a vertex and solution set.
Which of the following graphs shows a parabola with a vertex of $(1,-9)$ and solutions of $(-2,0)$ and $(4,0)$?
1. **State the problem:** We need to find the equation of a parabola with vertex at $(1,-9)$ and roots (solutions) at $(-2,0)$ and $(4,0)$.
2. **Recall the vertex form of a parabola:** $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex.
3. **Recall the factored form of a quadratic:** $$y = a(x - r_1)(x - r_2)$$ where $r_1$ and $r_2$ are the roots.
4. **Use the roots to write the factored form:** $$y = a(x + 2)(x - 4)$$
5. **Use the vertex to find $a$:** Substitute $x=1$, $y=-9$ into the factored form:
$$-9 = a(1 + 2)(1 - 4)$$
$$-9 = a(3)(-3)$$
$$-9 = -9a$$
6. **Solve for $a$:**
$$a = \frac{-9}{-9} = 1$$
7. **Write the full equation:**
$$y = (x + 2)(x - 4)$$
8. **Expand to standard form (optional):**
$$y = x^2 - 4x + 2x - 8 = x^2 - 2x - 8$$
9. **Check vertex from standard form:**
Vertex $x$-coordinate is $$x = -\frac{b}{2a} = -\frac{-2}{2 \times 1} = 1$$
Substitute $x=1$:
$$y = 1^2 - 2(1) - 8 = 1 - 2 - 8 = -9$$
This matches the given vertex.
**Final answer:** The parabola with vertex $(1,-9)$ and roots $(-2,0)$ and $(4,0)$ is $$y = (x + 2)(x - 4)$$ or $$y = x^2 - 2x - 8$$.
This parabola opens upwards and passes through the given points.