parabola | mrmillermath - Free Printable

Let’s go through each clue one by one and match it to the correct parabola graphs (A–I). We’ll look at key features: direction (up or down), vertex, axis of symmetry, zeros, range, domain, max/min values.

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Clue 1: My axis of symmetry is x = 2.

Axis of symmetry is the vertical line that splits the parabola in half. For a parabola, it passes through the vertex.

Look for graphs where the vertex is on x = 2.

- Graph A: vertex at x=2 → ✔
- Graph B: vertex at x=0 → ✘
- Graph C: vertex at x=0 → ✘
- Graph D: vertex at x=2 → ✔
- Graph E: vertex at x=2 → ✔
- Graph F: vertex at x=0 → ✘
- Graph G: vertex at x=1 → ✘
- Graph H: vertex at x=-1 → ✘
- Graph I: vertex at x=2 → ✔

So possible answers: A, D, E, I

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Clue 2: In my equation, a > 0.

“a” is the coefficient of x² in y = ax² + bx + c. If a > 0, parabola opens upward.

Which graphs open upward?

- A: opens down → ✘
- B: opens up → ✔
- C: opens up → ✔
- D: opens down → ✘
- E: opens up → ✔
- F: opens down → ✘
- G: opens up → ✔
- H: opens up → ✔
- I: opens down → ✘

Possible answers: B, C, E, G, H

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Clue 3: I have no zeros.

Zeros are where the graph crosses the x-axis (y=0). No zeros means the entire graph is above or below the x-axis and never touches it.

Check each:

- A: touches x-axis at two points → has zeros → ✘
- B: crosses x-axis twice → ✘
- C: vertex above x-axis, opens up → never touches x-axis → ✔
- D: crosses x-axis twice → ✘
- E: vertex above x-axis, opens up → never touches → ✔
- F: crosses x-axis twice → ✘
- G: crosses x-axis twice → ✘
- H: crosses x-axis twice → ✘
- I: crosses x-axis twice → ✘

Only C and E have no zeros.

Wait — let’s double-check E: vertex is at (2, 2) approximately? Looking at grid, yes, lowest point is above x-axis → no zeros → ✔
C: vertex at (0, 2) or so → also above → ✔

So: C, E

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Clue 4: My range is y ≤ 0.

Range is all possible y-values. y ≤ 0 means the parabola opens downward and its highest point (vertex) is at y=0 or lower.

Which graphs open down AND have max y-value ≤ 0?

- A: opens down, vertex at y=1 → range y≤1 → ✘
- D: opens down, vertex at y=0 → range y≤0 → ✔
- F: opens down, vertex at y=3 → ✘
- I: opens down, vertex at y=2 → ✘

Only D fits.

Wait — check D again: vertex is at (2, 0)? Yes, looks like it touches x-axis at peak → so max y=0 → range y≤0 → ✔

Any others? What about if vertex is below x-axis? None of the downward-opening ones have vertex below x-axis except maybe... no, all others are above.

So only D

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Clue 5: My zeros are at x = -2, and x = 4.

Zeros at x=-2 and x=4 → graph crosses x-axis at those points.

Find which graph has x-intercepts at -2 and 4.

Look at graph H: left intercept at x=-2, right at x=4? Let’s see:

Graph H: crosses x-axis at x=-2 and x=4? From grid: yes, appears to cross at (-2,0) and (4,0) → ✔

Any others? Graph B: crosses at x=-1 and x=3? Not matching.

Graph G: crosses at x=0 and x=2? No.

So only H

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Clue 6: My maximum value is y = 4.

Maximum value → parabola opens downward, and vertex y-coordinate is 4.

Which graph has vertex at y=4 and opens down?

Look at F: vertex at (0,4) → opens down → max y=4 → ✔

Any others? A: max y=1 → ✘
D: max y=0 → ✘
I: max y=2 → ✘

Only F

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Clue 7: I have a minimum value.

Minimum value → parabola opens upward (since it goes up on both sides, lowest point is vertex).

All upward-opening parabolas have a minimum.

From earlier: B, C, E, G, H open up → all have minimum.

But wait — does “have a minimum value” mean anything else? No, just that it opens up.

So: B, C, E, G, H

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Clue 8: My range is y ≥ -3.

Range y ≥ -3 → parabola opens upward, and vertex y-coordinate is -3.

Which graph has vertex at y=-3 and opens up?

Look at G: vertex at (1, -3)? From grid, yes — lowest point is at y=-3 → range y≥-3 → ✔

Any others? B: vertex at y=-2 → range y≥-2 → ✘
C: vertex at y=2 → ✘
E: vertex at y=2 → ✘
H: vertex at y=-2 → ✘

Only G

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Clue 9: My domain is all real numbers.

Domain is all x-values the graph covers. For any parabola (function), domain is always all real numbers — unless restricted, but none are here.

So ALL graphs A through I have domain all real numbers.

But since the question says “identify all possible answers”, we list all.

Answer: A, B, C, D, E, F, G, H, I

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Now, let’s compile the final answers per clue:

1. Axis of symmetry x=2 → A, D, E, I
2. a > 0 (opens up) → B, C, E, G, H
3. No zeros → C, E
4. Range y ≤ 0 → D
5. Zeros at x=-2 and x=4 → H
6. Maximum value y=4 → F
7. Have a minimum value → B, C, E, G, H
8. Range y ≥ -3 → G
9. Domain all real numbers → A, B, C, D, E, F, G, H, I

Final Answer:
1. A, D, E, I
2. B, C, E, G, H
3. C, E
4. D
5. H
6. F
7. B, C, E, G, H
8. G
9. A, B, C, D, E, F, G, H, I