Vertex Form of a Quadratic Equation | airSlate SignNow - Free Printable

Let’s solve each graph one by one. We’re looking for the vertex form of a quadratic equation:

> y = a(x - h)² + k

Where:
- (h, k) is the vertex of the parabola.
- “a” tells us how wide/narrow and which direction it opens (up or down).
- If a > 0 → opens up
- If a < 0 → opens down
- |a| > 1 → narrower than y = x²
- |a| < 1 → wider than y = x²

We’ll find the vertex first from the graph, then pick another point to solve for “a”.

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Problem 1:

Graph shows a parabola opening up, vertex at (2, 4).

So:
→ h = 2, k = 4 → y = a(x - 2)² + 4

Now pick another point on the graph. Let’s use (3, 5):

Plug in x=3, y=5:

5 = a(3 - 2)² + 4
5 = a(1) + 4
5 = a + 4
→ a = 1

✔ So equation: y = (x - 2)² + 4

Check with another point? Try (1, 5):

y = (1 - 2)² + 4 = (-1)² + 4 = 1 + 4 = 5 → correct.

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Problem 2:

Vertex at (4, 4), opens up.

Equation: y = a(x - 4)² + 4

Pick point (5, 5):

5 = a(5 - 4)² + 4
5 = a(1) + 4
→ a = 1

Check (3, 5):
y = (3 - 4)² + 4 = 1 + 4 = 5 → good.

✔ Equation: y = (x - 4)² + 4

Wait — let me double-check the graph. Actually, looking again — if vertex is at (4,4), and it goes through (5,5), yes, that’s slope 1, so a=1.

But wait — maybe I misread? Let me check problem 2 again.

Actually, in problem 2, the vertex looks like it's at (4, 4), and when x=5, y=5; x=3, y=5 — same as problem 1 but shifted right.

Yes, so same shape → a=1.

✔ Final: y = (x - 4)² + 4

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Problem 3:

Vertex at (-2, -6), opens up.

Equation: y = a(x + 2)² - 6

Pick point (0, -2):

-2 = a(0 + 2)² - 6
-2 = a(4) - 6
Add 6 to both sides:
4 = 4a
→ a = 1

Check with (1, 3)? Wait, let’s see what y should be at x=1:

y = (1+2)² - 6 = 9 - 6 = 3 → does graph show (1,3)? Looking at grid — yes, seems so.

Also check x=-1: y = (-1+2)² -6 = 1 -6 = -5 → graph shows around there? Yes.

✔ Equation: y = (x + 2)² - 6

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Problem 4:

Vertex at (-4, 2), opens up.

Equation: y = a(x + 4)² + 2

Pick point (-3, 3):

3 = a(-3 + 4)² + 2
3 = a(1) + 2
→ a = 1

Check (-5, 3):
y = (-5 + 4)² + 2 = 1 + 2 = 3 → matches.

✔ Equation: y = (x + 4)² + 2

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Problem 5:

This one opens down! Vertex at (0, -2)

So: y = a(x - 0)² - 2 → y = ax² - 2

Pick point (2, -6):

-6 = a(2)² - 2
-6 = 4a - 2
Add 2: -4 = 4a
→ a = -1

Check (-2, -6):
y = -1*(-2)^2 -2 = -4 -2 = -6 → good.

Also check (1, -3):
y = -1*(1)^2 -2 = -1 -2 = -3 → matches graph.

✔ Equation: y = -x² - 2

Or written as: y = -(x - 0)² - 2

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Problem 6:

Vertex at (4, -4), opens up.

Equation: y = a(x - 4)² - 4

Pick point (5, -3):

-3 = a(5 - 4)² - 4
-3 = a(1) - 4
Add 4: 1 = a
→ a = 1

Check (3, -3):
y = (3-4)² -4 = 1 -4 = -3 → good.

Check (6, 0):
y = (6-4)² -4 = 4 -4 = 0 → matches graph.

✔ Equation: y = (x - 4)² - 4

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## ✔ Final Answers:

1. y = (x - 2)² + 4
2. y = (x - 4)² + 4
3. y = (x + 2)² - 6
4. y = (x + 4)² + 2
5. y = -x² - 2
6. y = (x - 4)² - 4