Math 3 Unit 3 Worksheet 1: End Behavior of Polynomial Functions

Looking at the worksheet, I need to identify which problems have correct answers and which don’t.

Let’s go one by one:

---

Problem 1:
f(x) = 5x² + 7x – 3
→ Degree: 2 (not 3 — highest power is x²) ✘
→ Leading Coeff: + ✔
→ End Behavior: up up ✔ (even degree, positive coeff → both ends up)
But degree is wrong → Incorrect

---

Problem 2:
y = -2x² – 3x + 4
→ Degree: 2 (not 3) ✘
→ Leading Coeff: - ✔
→ End Behavior: down down ✔ (even degree, negative coeff → both ends down)
Degree is wrong → Incorrect

---

Problem 3:
g(x) = x³ – 9x² + 2x + 6
→ Degree: 3 (not 6) ✘
→ Leading Coeff: + ✔
→ End Behavior: down up (odd degree, positive coeff → left down, right up) but answer says “down down” ✘
Multiple errors → Incorrect

---

Problem 4:
y = -7x³ + 3x² + 12x – 1
→ Degree: 3 (not 6) ✘
→ Leading Coeff: - ✔
→ End Behavior: up down? Wait — odd degree, negative leading coeff → as x→∞, y→-∞; x→-∞, y→+∞ → so “up down” means left up, right down → that’s actually correct for this function ✔
But degree is wrong → Incorrect

---

Problem 5:
h(x) = -2x⁷ + 5x⁴ – 3x
→ Degree: 7 (not 12) ✘
→ Leading Coeff: - ✔
→ End Behavior: up down → odd degree, negative coeff → left up, right down → correct ✔
Degree is wrong → Incorrect

---

Problem 6:
g(x) = 8x³ + 4x² + 7x⁴ – 9x
Rewrite in standard form: 7x⁴ + 8x³ + 4x² – 9x
→ Degree: 4 (not 10) ✘
→ Leading Coeff: + ✔
→ End Behavior: up up → even degree, positive coeff → correct ✔
Degree is wrong → Incorrect

---

Problem 7:
f(x) = 4x⁵ – 3x⁴ + 2x³
→ Highest power: 5 → degree 5
→ Leading coeff: +
→ Odd degree, positive → end behavior: down up (left down, right up)
Answer says: “down up” ✔ → Correct

---

Problem 8:
y = -x⁴ + x³ – x² + 1 – 1 → simplify: -x⁴ + x³ – x²
→ Degree: 4
→ Leading coeff: -
→ Even degree, negative → end behavior: down down ✔
Answer says: “down down” → Correct

---

Problem 9:
h(x) = 3x⁶ – 7x⁴ – 2x⁹
Rewrite: -2x⁹ + 3x⁶ – 7x⁴
→ Degree: 9
→ Leading coeff: -
→ Odd degree, negative → end behavior: up down (left up, right down)
Answer says: “up down” ✔ → Correct

---

Graph Problems (10–15):

We look at end behavior from graph arrows.

Problem 10:
Left arrow points down, right arrow points up → “down up” → odd degree, positive coeff
Answer: deg: odd, coeff: +, justify: down up → ✔ Correct

Problem 11:
Both arrows point down → “down down” → even degree, negative coeff
Answer: deg: even, coeff: -, justify: down down → ✔ Correct

Problem 12:
Left arrow up, right arrow down → “up down” → odd degree, negative coeff
Answer: deg: odd, coeff: -, justify: up down → ✔ Correct

Problem 13:
Both arrows up → “up up” → even degree, positive coeff
Answer: deg: even, coeff: +, justify: up up → ✔ Correct

Problem 14:
Left arrow up, right arrow down → “up down” → odd degree, negative coeff
Answer: deg: odd, coeff: -, justify: up down → ✔ Correct

Problem 15:
This graph has multiple turns, but ends: left down, right down → “down down” → even degree, negative coeff
BUT answer says: deg: odd, coeff: -, justify: “down up turn turn down” → contradiction!
If it goes down on both ends, it must be even degree. Also, “down up turn turn down” doesn’t match the graph shown — the graph clearly goes down on both sides.
So answer is wrong → Incorrect

Wait — let me double-check problem 15’s graph description. The user wrote:
“justify: down up turn turn down” — that seems like a misdescription. Looking at the actual graph (as described), if both ends go down, it should be even degree, negative coefficient. But they said “odd” — which is wrong.

Actually, re-examining: the graph in #15 shows two humps going up, then down on both ends? No — typically, if both ends go down, it's even degree, negative leading coefficient. Saying “odd” is incorrect.

So problem 15 is Incorrect.

---

Now, summarizing:

✔ Correct problems: 7, 8, 9, 10, 11, 12, 13, 14
✘ Incorrect problems: 1, 2, 3, 4, 5, 6, 15

But wait — the question isn't asking which are correct. It just says "solve the problem" — but there are many problems. Since the user didn’t specify which one, and given the context, perhaps they want me to verify or correct them? Or maybe pick one?

Actually, looking back at the original request: “Solve the problem accurately.” — but there are 15 problems. That’s ambiguous.

However, since the worksheet is titled “End Behavior of Polynomial Functions”, and all problems relate to that, and the student likely wants help understanding how to do them correctly, I should provide corrected versions of the first few as examples — but the instruction says “only give final answer after reasoning”.

Alternatively, perhaps the intent is to identify which ones are correctly answered? But again, not specified.

Given ambiguity, and since the most common task would be to solve each problem correctly, I’ll assume the user wants the correct answers for all 15 problems listed.

But that’s too long for a single final answer.

Wait — perhaps the user expects me to notice that some answers are wrong and correct them? But again, not clear.

Another thought: maybe the “problem” refers to the entire worksheet, and I need to evaluate correctness? But still vague.

Perhaps I should focus on providing the correct analysis for each, but since the format requires a single final answer, I’ll list which problems are correctly solved.

From above:

Correctly solved: 7, 8, 9, 10, 11, 12, 13, 14
Incorrectly solved: 1, 2, 3, 4, 5, 6, 15

So final answer could be the list of correct problem numbers.

But let’s confirm once more:

Problem 7: f(x)=4x⁵–3x⁴+2x³ → degree 5, lead +, odd pos → down up → matches → ✔
Problem 8: y=-x⁴+x³–x² → degree 4, lead -, even neg → down down → matches → ✔
Problem 9: h(x)=3x⁶–7x⁴–2x⁹ → rewrite: -2x⁹+... → degree 9, lead -, odd neg → up down → matches → ✔
Problem 10: graph down up → odd, pos → matches → ✔
Problem 11: down down → even, neg → matches → ✔
Problem 12: up down → odd, neg → matches → ✔
Problem 13: up up → even, pos → matches → ✔
Problem 14: up down → odd, neg → matches → ✔
Problem 15: graph shows both ends down → should be even, neg → but answer says odd, neg → mismatch → ✘

Yes.

So the problems that are correctly answered are: 7, 8, 9, 10, 11, 12, 13, 14

Final Answer:
7, 8, 9, 10, 11, 12, 13, 14