SOLUTION: Kami Export - M3 U3 WS1 - End Behavior of Polynomial ... - Free Printable

Let’s go through each problem one by one, step by step.

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Problem 16:
We need a polynomial where:
- As x → -∞ (left), f(x) → +∞
- As x → +∞ (right), f(x) → -∞

This means the graph goes up on the left and down on the right.

That happens when:
→ The degree is odd (so ends go in opposite directions)
→ The leading coefficient is negative (because for odd degrees, negative lead coeff = up-left, down-right)

Example:
Try a cubic (degree 3) with negative leading term.

Given answer: `-12x^3 + 9x + 4`

Check:
- Degree = 3 → odd ✔
- Leading coefficient = -12 → negative ✔
So as x → -∞, (-12)(very large negative)^3 = (-12)(very large negative) = very large positive → +∞ ✔
As x → +∞, (-12)(very large positive)^3 = very large negative → -∞ ✔

✔️ This works.

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Problem 17:
End behavior:
- Left: f(x) → +∞
- Right: f(x) → +∞

Graph goes up on both sides.

That happens when:
→ Degree is even (ends go same direction)
→ Leading coefficient is positive

Given answer: `3x^2 - 13 + 7` → simplify to `3x^2 - 6`

Check:
- Degree = 2 → even ✔
- Leading coefficient = 3 → positive ✔
As x → ±∞, 3x² dominates → always positive → +∞ on both sides ✔

✔️ This works.

(Note: The original expression “3x^2 -13 +7” simplifies to “3x^2 -6”, which is fine.)

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Problem 18:
Sketch a graph of a polynomial with:
→ Negative lead coefficient
→ Even degree

Even degree + negative lead coeff → graph opens downward on both ends.

Think of an upside-down U shape — like a frown.

Example: y = -x² or y = -x⁴

Sketch idea:
- Draw axes.
- Curve starts high on left, comes down, maybe has a bump or dip, then goes back down on right.
- Both ends point downward.

You don’t need exact points — just show the end behavior correctly.

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Problem 19:
Sketch a graph of a polynomial with:
→ Positive lead coefficient
→ Odd degree

Odd degree + positive lead coeff → graph goes:
- Down on the left (as x → -∞, f(x) → -∞)
- Up on the right (as x → +∞, f(x) → +∞)

Like a sideways S that rises from bottom-left to top-right.

Example: y = x³ or y = x⁵

Sketch idea:
- Start low on left, curve up through origin or somewhere, keep going up on right.
- May have wiggles, but overall trend is rising left to right.

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Problem 20:
Equation given:
f(x) = x⁴ + x³ - 2x² - 1

We are told this matches the graph shown (which we can’t see, but we can analyze the function).

Let’s check key features:

Degree = 4 → even → ends go same direction
Leading coefficient = +1 → positive → so both ends go to +∞

Now let’s find some points to sketch or verify:

At x = 0:
f(0) = 0 + 0 - 0 - 1 = -1 → point (0, -1)

At x = 1:
f(1) = 1 + 1 - 2 - 1 = -1 → (1, -1)

At x = -1:
f(-1) = 1 -1 -2 -1 = -3 → (-1, -3)

At x = 2:
f(2) = 16 + 8 - 8 -1 = 15 → (2, 15)

At x = -2:
f(-2) = 16 -8 -8 -1 = -1 → (-2, -1)

So it dips below zero around x=-1 to x=1, then shoots up on both sides.

The graph should look like a W-shape but flattened at bottom? Actually, since it's degree 4 with positive lead, it should be U-shaped overall, but may have local max/min.

Actually, let’s factor if possible? Not easily. But based on values:

It crosses y-axis at -1, goes down to about -3 at x=-1, back to -1 at x=0 and x=1, then up sharply after x=1.

So the graph likely has two turning points — typical for quartic.

Since the problem says “the equation... to the right is” and gives this formula, and assuming the graph matches, then this is correct.

But wait — the user didn’t ask to verify, they just gave the equation. So perhaps the task was to write the equation based on the graph? But since no graph is visible to us, and the problem states the equation, I think we’re meant to accept it as given.

However, looking back — Problem 20 says: “The equation of the polynomial function to the right is...” followed by the formula. So probably the student is supposed to match or confirm — but without seeing the graph, we can’t do more.

But since the instruction is to solve the problem accurately, and the only thing asked is to state the equation (which is already given), perhaps there’s nothing to compute here.

Wait — actually, re-reading: the problem says “The equation... is” and then writes the formula. So maybe the task was to identify or write it — but it’s already written.

Perhaps in the original worksheet, the graph was shown and student had to pick/write the equation. Since we don’t have the graph, and the equation is provided, we’ll assume it’s correct as stated.

But let me double-check if this function could produce a graph that looks like what might be expected.

We know:
- Ends go up (even degree, positive lead) ✔
- Has real roots? Let’s try to find zeros.

Set f(x) = x⁴ + x³ - 2x² - 1 = 0

Try rational root theorem: possible roots ±1

f(1) = 1+1-2-1 = -1 ≠ 0
f(-1) = 1 -1 -2 -1 = -3 ≠ 0
No rational roots obvious.

But that’s okay — doesn’t need to cross x-axis.

From earlier calculations:
- At x=-2: f=-1
- x=-1: f=-3
- x=0: f=-1
- x=1: f=-1
- x=2: f=15

So between x=1 and x=2, it goes from -1 to 15 → must cross x-axis once there.

Similarly, maybe another crossing? Let’s check x=1.5:

f(1.5) = (1.5)^4 + (1.5)^3 - 2*(1.5)^2 -1
= 5.0625 + 3.375 - 4.5 -1 = (8.4375) - 5.5 = 2.9375 >0

So between x=1 and 1.5, it crosses from -1 to +2.9 → root near 1.2 or so.

What about left side? From x=-2 to x=-1: f(-2)=-1, f(-1)=-3 → still negative.

x=-3: f(-3)=81 -27 -18 -1= 35>0 → ah! So between x=-3 and x=-2, it goes from +35 to -1 → crosses x-axis there too.

So two real roots: one between -3 and -2, one between 1 and 1.5.

And since degree 4, possibly two more complex roots.

Graph would come down from +∞ on left, cross x-axis around x≈-2.5, go down to min around x=-1, rise a bit but stay negative until x≈1.2, cross again, then go up to +∞.

So shape: starts high left, dips below axis, comes up slightly but stays under, then crosses up and goes high right.

If the graph in the image shows that, then equation is correct.

Since we can't see the graph, and the problem presents the equation as fact, we'll take it as given.

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Now, compiling final answers clearly:

Final Answer:
16. -12x³ + 9x + 4
17. 3x² - 6 (or simplified form of given expression)
18. Sketch: Graph opening downward on both ends (like an inverted U or frown shape)
19. Sketch: Graph going from bottom-left to top-right (rising overall, like a stretched S starting low and ending high)
20. f(x) = x⁴ + x³ - 2x² - 1