Question

We can use an algebraic method or a Desmos method to write the equation of a parabola given a graph in vertex form if we know the vertex, and a couple other points on the graph.

`y=a\left(x-h\right)^{2}+k`

*If we are given equations in multiple choice format, you could use Desmos to help you choose, but be sure to check multiple features on the graph, not just the vertex!


A. y = 2(x - 5)^2 + 3
B. y = 2(x - 5)^2 - 3
C. y = 4(x - 5)^2 - 3
D. y= 4(x - 5)^2 - 3


Algebraic Method:

1. Name the Vertex `(h,\ k)`.

2. Name another point `(x,\ y)` on the graph - avoid decimals.

3. Substitute `x,\ y\ \&\ h,\ k` into the equation to find "`a`".

4. Substitute `a,\ h,\ k` back into the equation to write it in vertex form.

Answer 1

I can’t pick the correct choice until I know the vertex and at least one other point on the graph (or see the graph). Here’s how to decide quickly and how to test the four choices you listed.

Quick checks
- First check the vertex (h,k). Any candidate equation must have the same h and k.
- Next pick one other point on the parabola with integer coordinates (avoid decimals), call it (x,y). Substitute into y = a(x − h)^2 + k and solve for a.
- Or (for multiple choice) substitute that (x,y) into each candidate and see which one is true.

Notes about your choices
- A: y = 2(x − 5)^2 + 3 → vertex (5, 3), a = 2
- B: y = 2(x − 5)^2 − 3 → vertex (5, −3), a = 2
- C: y = 4(x − 5)^2 − 3 → vertex (5, −3), a = 4
- D: y = 4(x − 5)^2 − 3 → same as C (duplicate)

Example (to show the algebraic step)
- Suppose the vertex from the graph is (5, −3) and another point on the parabola is (6, 1).
- Substitute (6,1): 1 = a(6−5)^2 + (−3) → 1 = a(1) − 3 → a = 4.
- So the correct choice would be C (or D, since it’s identical to C).

Desmos method
- Plot the four equations and the given graph; the matching equation will lie exactly on the plotted parabola. But don’t rely only on the vertex — also check one or two other visible points (intercepts or another lattice point).

If you paste the graph or give me the vertex and one other point (an easy integer point), I’ll pick the correct choice for you.