Question: 2 — Converting a Quadratic Equation into Vertex Form
LEARNING OBJECTIVE: Write a quadratic equation in vertex form.
The equation of the parabola $y = 2x^2 - 8x + 13$ in vertex form is
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a.) $y = 2(x + 2)^2 + 5$
b.) $y = 2(x + 2)^2 - 5$
c.) $y = 2(x - 2)^2 - 5$
d.) $y = 2(x - 2)^2 + 5"
1. **State the problem:** Convert the quadratic equation $y = 2x^2 - 8x + 13$ into vertex form.
2. **Recall the vertex form of a quadratic:** $$y = a(x - h)^2 + k$$ where $(h,k)$ is the vertex.
3. **Start with the given equation:** $$y = 2x^2 - 8x + 13$$
4. **Factor out the coefficient of $x^2$ from the first two terms:**
$$y = 2(x^2 - 4x) + 13$$
5. **Complete the square inside the parentheses:**
Take half of the coefficient of $x$, which is $-4$, half is $-2$, square it to get $4$.
6. **Add and subtract $4$ inside the parentheses:**
$$y = 2(x^2 - 4x + 4 - 4) + 13$$
7. **Group the perfect square trinomial and simplify:**
$$y = 2((x - 2)^2 - 4) + 13$$
8. **Distribute the $2$:**
$$y = 2(x - 2)^2 - 8 + 13$$
9. **Combine constants:**
$$y = 2(x - 2)^2 + 5$$
10. **Final vertex form:** $$y = 2(x - 2)^2 + 5$$
This matches option d.)