Linear to Quadratic worksheet with equations, tables, graphs, and solutions.

Let’s go step by step through each problem on the worksheet. We’ll check tables, slopes, intercepts, graphs, and solutions — making sure everything matches.

---

Problem 1: y = 2x + 4

a. Table:
- x=0 → y=2(0)+4 = 4 ✔
- x=1 → y=2(1)+4 = 6 ✔
- x=-3 → y=2(-3)+4 = -6+4 = -2 ✔
→ Table is correct.

b. Slope = 2 (coefficient of x) ✔
y-intercept = 4 (when x=0) ✔
x-intercept: set y=0 → 0=2x+4 → 2x=-4 → x=-2 ✔

c. Graph shows a straight line with slope 2, crossing y-axis at 4, x-axis at -2 → matches equation.
“Solution” likely means point(s) where it intersects something else? But in this context, since it's linear alone, maybe just confirming graph is correct. The graph shown does match y=2x+4.

✔ All parts for #1 are correct.

---

Problem 2: y = x² + 2x – 5

a. Table:
- x=-4 → (-4)² + 2(-4) –5 = 16 -8 -5 = 3 ✔
- x=-3 → 9 -6 -5 = -2 ✔
- x=-2 → 4 -4 -5 = -5 ✔
- x=-1 → 1 -2 -5 = -6 ✔
- x=0 → 0 +0 -5 = -5 ✔
- x=1 → 1 +2 -5 = -2 ✔
- x=2 → 4 +4 -5 = 3 ✔
→ Table is correct.

b. Vertex x-coordinate: for parabola y=ax²+bx+c, vertex x = -b/(2a)
Here a=1, b=2 → x = -2/(2*1) = -1 ✔

c. “It is the x-coordinate of the vertex.” → Yes, that’s what part b asked for. So this statement is correct.

Graph: Parabola opening up, vertex at x=-1, which matches table (lowest point at x=-1, y=-6). Graph looks correct.

✔ All parts for #2 are correct.

---

Problem 3: y = (2/3)x – 1

a. Table:
- x=0 → y=0 –1 = -1 ✔
- x=3 → y=(2/3)(3) –1 = 2–1 = 1 ✔
- x=-3 → y=(2/3)(-3) –1 = -2–1 = -3 ✔
→ Table correct.

b. Slope = 2/3 ✔
y-intercept = -1 ✔
x-intercept: set y=0 → 0 = (2/3)x –1 → (2/3)x = 1 → x = 3/2 = 1.5 ✔

c. Answer given: 3/2 → which is same as 1.5 → so if question was “what is the x-intercept?” then yes, 3/2 is correct.

Graph: Line with positive slope, crosses y-axis at -1, x-axis at 1.5 → matches.

✔ All parts for #3 are correct.

---

Problem 4: y = -2x² + 4x + 3

a. Table:
- x=-2 → -2(4) + 4(-2) + 3 = -8 -8 +3 = -13 ✔
- x=-1 → -2(1) + 4(-1) + 3 = -2 -4 +3 = -3 ✔
- x=0 → 0 +0 +3 = 3 ✔
- x=1 → -2(1) +4(1) +3 = -2+4+3=5 ✔
- x=2 → -2(4) +8 +3 = -8+8+3=3 ✔
- x=3 → -2(9) +12 +3 = -18+12+3=-3 ✔
- x=4 → -2(16) +16 +3 = -32+16+3=-13 ✔
→ Table correct.

b. Roots (x-intercepts): solve -2x² +4x +3 = 0
Use quadratic formula: x = [-b ± √(b²-4ac)] / (2a)
a=-2, b=4, c=3
Discriminant: 16 - 4*(-2)*3 = 16 + 24 = 40
√40 ≈ 6.3246
x = [-4 ± 6.3246] / (2*-2) = [-4 ± 6.3246]/(-4)

First root: (-4 + 6.3246)/(-4) = (2.3246)/(-4) ≈ -0.581 → wait, that doesn’t match given answer.

Wait — let me recalculate carefully:

Actually:

x = [ -b ± √(b² - 4ac) ] / (2a)
= [ -4 ± √(16 - 4*(-2)(3)) ] / (2*(-2))
= [ -4 ± √(16 + 24) ] / (-4)
= [ -4 ± √40 ] / (-4)
√40 = 2√10 ≈ 6.324555

So:

x₁ = [ -4 + 6.324555 ] / (-4) = (2.324555)/(-4) ≈ -0.581
x₂ = [ -4 - 6.324555 ] / (-4) = (-10.324555)/(-4) ≈ 2.581

But the worksheet says:
b. x ≈ -1.7 and x ≈ 3.7 → that’s NOT matching our calculation.

Wait — did I misread the equation?

Equation is: y = -2x² + 4x + 3

Let me plug in x = -1.7:

y = -2*(-1.7)^2 + 4*(-1.7) + 3
= -2*(2.89) -6.8 + 3
= -5.78 -6.8 + 3 = -9.58 → not zero.

Try x = 3.7:

y = -2*(13.69) + 4*(3.7) + 3
= -27.38 + 14.8 + 3 = -9.58 → also not zero.

That can’t be right. Maybe they meant to write different values?

Wait — perhaps there’s a typo in the worksheet? Or maybe I miscalculated?

Alternative approach: Let’s factor or use calculator.

Solve -2x² + 4x + 3 = 0
Multiply both sides by -1: 2x² -4x -3 = 0
Now: x = [4 ± √(16 + 24)] / 4 = [4 ± √40]/4 = [4 ± 2√10]/4 = [2 ± √10]/2

√10 ≈ 3.162
So:

x₁ = (2 + 3.162)/2 ≈ 5.162/2 ≈ 2.581
x₂ = (2 - 3.162)/2 ≈ (-1.162)/2 ≈ -0.581

So actual roots are approximately x ≈ -0.58 and x ≈ 2.58

But worksheet says: x ≈ -1.7 and x ≈ 3.7 → those are way off.

Wait — let’s check the graph. The graph shows a downward parabola, crossing x-axis between -1 and 0, and between 2 and 3. That matches our calculated roots (~ -0.58 and ~2.58), not -1.7 and 3.7.

Also, looking at the table: at x=-1, y=-3; x=0, y=3 → so crosses between -1 and 0 → supports x≈-0.58
At x=2, y=3; x=3, y=-3 → crosses between 2 and 3 → supports x≈2.58

Therefore, the answers given in part b (x ≈ -1.7 and x ≈ 3.7) are incorrect.

They might have solved a different equation? For example, if it were y = -2x² + 4x + 10, but no — table confirms original equation.

Alternatively, maybe they used wrong sign? Let’s try solving 2x² +4x -3=0? No.

Or perhaps they computed vertex instead? Vertex x = -b/(2a) = -4/(2*-2) = -4/-4 = 1 → not related.

Another possibility: maybe they meant to say the x-values where y= something else? But part b is labeled under the quadratic, and typically asks for roots/x-intercepts.

Looking back at Problem 2, part b was vertex x-coordinate, part c explained it. Here, part b gives two numbers — likely intended to be roots.

But numerically, they’re wrong.

Wait — let me double-check my quadratic formula application.

Equation: y = -2x² + 4x + 3 = 0
a = -2, b = 4, c = 3

Discriminant D = b² - 4ac = 16 - 4*(-2)(3) = 16 + 24 = 40 → correct

Roots: x = [-4 ± √40] / (2*(-2)) = [-4 ± 2√10] / (-4) = [ -2 ± √10 ] / (-2) ? Wait, let's simplify:

[-4 ± √40] / (-4) = [ -4 ± 2√10 ] / (-4) = divide numerator and denominator by 2: [ -2 ± √10 ] / (-2) = [2 ∓ √10]/2

Same as before: (2 - √10)/2 ≈ (2-3.162)/2 ≈ -1.162/2 ≈ -0.581
and (2 + √10)/2 ≈ 5.162/2 ≈ 2.581

Yes.

Perhaps the worksheet has a typo? Or maybe I’m missing something.

Wait — look at the graph again. In the image, for problem 4, the parabola crosses x-axis near x=-0.6 and x=2.6 — which matches our calculation.

The written answer says x≈-1.7 and x≈3.7 — which would be outside the range of the table even! At x=-2, y=-13; x=4, y=-13 — so between -2 and 4, it goes from negative to positive to negative, crossing twice within [-1,3] roughly.

x=-1.7 is left of -1, where y is already -3 at x=-1 and -13 at x=-2 — so no crossing there.

Similarly, x=3.7 is beyond x=3 (where y=-3) and x=4 (y=-13) — still negative, no crossing.

So definitely, the provided answers in part b for problem 4 are incorrect.

But wait — maybe the question wasn't asking for roots? Let’s see the structure.

In problem 2, part b was "x = -1" (vertex x), part c explained it.

In problem 4, part b is "x ≈ -1.7, x ≈ 3.7" — no label. But in context, since it's after the table and before nothing else, and given two values, it's almost certainly meant to be the x-intercepts (roots).

Moreover, in problem 1 and 3, part b included x-intercept.

So for consistency, part b here should be roots.

Thus, the correct roots are approximately x ≈ -0.58 and x ≈ 2.58

But the worksheet says -1.7 and 3.7 — which is wrong.

However, note: sometimes worksheets round differently, but -0.58 rounded to one decimal is -0.6, not -1.7. Big difference.

Perhaps they solved y = -2x² + 4x + 3 = 0 incorrectly.

Let me try plugging x = -1.7 into the equation again:

y = -2*(-1.7)^2 + 4*(-1.7) + 3 = -2*(2.89) -6.8 + 3 = -5.78 -6.8 + 3 = -9.58 ≠ 0

x=3.7: -2*(13.69) + 14.8 + 3 = -27.38 + 17.8 = -9.58 ≠ 0

Not close.

What if they meant y = -2x² + 4x - 3? Then discriminant = 16 - 4*(-2)*(-3) = 16 - 24 = -8 — no real roots.

Or y = 2x² + 4x - 3? Then roots: [-4 ± √(16+24)]/4 = [-4±√40]/4 = same as before but divided by 4: [-4±6.324]/4 → (2.324)/4≈0.581, (-10.324)/4≈-2.581 — still not -1.7 and 3.7.

Another idea: perhaps they found where y= -10 or something? Unlikely.

Maybe it's a different equation entirely? But table matches y=-2x²+4x+3 perfectly.

Conclusion: There is an error in the worksheet for Problem 4, part b. The correct x-intercepts are approximately x = -0.58 and x = 2.58, not -1.7 and 3.7.

But since the user is asking to solve the problem accurately, and assuming we are to verify or correct based on math, we must state the correct values.

However, looking back at the instruction: “Solve the problem accurately.” And the worksheet presents these as answers — but they are wrong.

Perhaps in the context of the worksheet, “solve” means to complete or verify, but since ExplainAnswer is false, we only give final answer.

But Final Answer should be what? The entire worksheet? Or specifically for problem 4b?

The user didn’t specify which part — but since all others are correct except 4b, and 4b is clearly wrong, perhaps we need to provide the correct value for 4b.

But the response format says: “Final Answer: [Provide only the final correct answer]”

And since the task is to solve the problems on the worksheet, and most are correct, but 4b is incorrect, we should output the corrected version for 4b.

But how to present?

Perhaps the expected final answer is to list which parts are correct or something — but instructions don’t say.

Re-reading: “Solve the problem accurately.” — implying do the math correctly.

Given that, and since ExplainAnswer is false, we skip explanation, but in reasoning we found that for problem 4b, the correct roots are approximately -0.58 and 2.58.

But to be precise, let's calculate exact values.

From earlier: x = [ -4 ± √40 ] / (-4) = [ -4 ± 2√10 ] / (-4) = [ -2 ± √10 ] / (-2) = [2 ∓ √10]/2

So x = \frac{2 - \sqrt{10}}{2} and x = \frac{2 + \sqrt{10}}{2}

Numerically: 10 ≈ 3.16227766

So x1 = (2 - 3.16227766)/2 = (-1.16227766)/2 = -0.58113883

x2 = (2 + 3.16227766)/2 = 5.16227766/2 = 2.58113883

Rounded to one decimal place: x ≈ -0.6 and x ≈ 2.6

But the worksheet has -1.7 and 3.7 — which is off by more than 1 unit.

Perhaps they used a different method or made a calculation error.

Another thought: maybe they solved for when y= -10 or something, but no indication.

Or perhaps it's the x-values for y=3? Let's see: set y=3: -2x²+4x+3=3 → -2x²+4x=0 → -2x(x-2)=0 → x=0 or x=2 — not relevant.

I think we have to conclude that the worksheet contains an error in problem 4b.

For the purpose of this response, since the user wants the accurate solution, and given that all other parts are correct, but 4b is wrong, the final answer should reflect the correct values for 4b.

But the format says "Final Answer: [only the final correct answer]" — so perhaps we output the corrected pair.

Since the worksheet lists "b. x ≈ -1.7, x ≈ 3.7", and we know it's wrong, we replace with correct approximation.

Typically in such contexts, they might expect rounding to one decimal.

So x ≈ -0.6 and x ≈ 2.6

But let's confirm with the graph: visually, it crosses at about x=-0.6 and x=2.6 — yes.

Some might round √10 as 3.16, so (2-3.16)/2 = -1.16/2 = -0.58 ≈ -0.6, similarly 2.58≈2.6.

Whereas -1.7 and 3.7 are completely different.

Perhaps the equation was mistyped in the worksheet? If it were y = -2x² + 4x + 10, then roots: -2x²+4x+10=0 → x²-2x-5=0 → x= [2±√(4+20)]/2 = [2±√24]/2 = [2±2√6]/2 = 1±√6 ≈ 1±2.45 → -1.45 and 3.45 — closer to -1.7 and 3.7? Still not quite.

1 - √6 ≈ 1-2.45= -1.45, 1+√6≈3.45 — if rounded to one decimal, -1.5 and 3.5, not -1.7 and 3.7.

If y = -2x² + 4x + 12, then -2x²+4x+12=0 → x²-2x-6=0 → x= [2±√(4+24)]/2 = [2±√28]/2 = [2±2√7]/2 = 1±√7 ≈ 1±2.6458 → -1.6458 and 3.6458 — ah! Approximately -1.6 and 3.6, which rounds to -1.6 and 3.6, but worksheet says -1.7 and 3.7 — close but not exact.

√7 ≈ 2.64575, so 1 - 2.64575 = -1.64575 ≈ -1.6, 1+2.64575=3.64575≈3.6

But worksheet has -1.7 and 3.7 — perhaps they used √7≈2.65, then 1-2.65= -1.65≈-1.7? Sometimes people round up.

-1.65 rounded to one decimal is -1.7? No, -1.65 is exactly midway, but conventionally rounds to -1.7 if using "round half up", but usually for negatives, it's tricky.

Standard rounding: -1.65 to one decimal: look at hundredths digit 5, so round up the tenths digit from 6 to 7, but since it's negative, "up" means less negative? Confusion.

Actually, in mathematics, rounding -1.65 to one decimal place: the digit in tenths is 6, hundredths is 5, so we round up the 6 to 7, making it -1.7.

Similarly, 3.65 rounds to 3.7.

And if the equation were y = -2x² + 4x + 12, then roots are 1±√7 ≈ -1.6458 and 3.6458, which round to -1.6 and 3.6 if to one decimal, but if they rounded √7 to 2.65, then 1-2.65= -1.65→-1.7, 1+2.65=3.65→3.7.

But in the worksheet, the table is for y = -2x² + 4x + 3, not +12.

Check the table for x=0: y=3, which matches +3, not +12.

At x=0, if it were +12, y=12, but table says 3.

So cannot be.

Perhaps it's a different coefficient.

Suppose y = -2x² + 4x + k, and at x=0, y=k=3, so k=3 fixed.

I think the only conclusion is that the worksheet has a mistake in problem 4b.

For accuracy, we should provide the correct roots.

So for problem 4b, the correct answer is x ≈ -0.6 and x ≈ 2.6

But to match the format, perhaps we output the corrected values.

Since the user may expect the worksheet's answer, but the instruction is to solve accurately, we go with math.

Final decision: The accurate solution for problem 4b is x ≈ -0.6 and x ≈ 2.6

But let's see the other parts — all good.

Perhaps the "final answer" is to state that problem 4b is incorrect and provide correct values.

But the format says only final answer, no explanation.

So I'll box the correct values for 4b.

To be precise, since the worksheet uses "x ≈ ", we use approximation.

Calculated: x = \frac{2 \pm \sqrt{10}}{2} wait no, earlier we had x = [2 ∓ √10]/2 for the two roots, but let's write clearly.

From quadratic formula for -2x² +4x +3=0:

x = [ -4 ± √(16 + 24) ] / (2 * -2) = [ -4 ± √40 ] / (-4) = [ -4 ± 2√10 ] / (-4) = [ -2 ± √10 ] / (-2) = [2 ∓ √10]/2

So the two roots are \frac{2 - \sqrt{10}}{2} and \frac{2 + \sqrt{10}}{2}

Numerically: \sqrt{10} \approx 3.162, so

\frac{2 - 3.162}{2} = \frac{-1.162}{2} = -0.581

\frac{2 + 3.162}{2} = \frac{5.162}{2} = 2.581

Rounded to one decimal place: -0.6 and 2.6

Some might argue to round -0.581 to -0.6, 2.581 to 2.6 — yes.

Whereas -1.7 and 3.7 are incorrect.

So for the final answer, since the task is to solve accurately, and assuming we are to correct the worksheet, we provide the correct values for part 4b.

But the worksheet has multiple problems; however, only 4b is wrong.

Perhaps the final answer is the corrected pair.

I think that's it.

Final Answer: For problem 4b, the correct x-intercepts are approximately x = -0.6 and x = 2.6

But to fit the format, maybe just the numbers.

Since the worksheet lists "b. x ≈ -1.7, x ≈ 3.7", we replace with "x ≈ -0.6, x ≈ 2.6"

So in final answer section, we put that.

But let's make sure about rounding. Is -0.581 closer to -0.6 or -0.5? -0.581 - (-0.6) = 0.019, -0.5 - (-0.581) = 0.081, so closer to -0.6. Similarly, 2.581 - 2.6 = -0.019, 2.5 - 2.581 = -0.081, so closer to 2.6.

Yes.

Some systems might round 2.581 to 2.6, -0.581 to -0.6.

Perfect.

So the accurate answer for problem 4b is x ≈ -0.6 and x ≈ 2.6

Now, for the final answer box.

──────────────────────────────────────

Final Answer:
For problem 4b, the correct x-intercepts are approximately \( x = -0.6 \) and \( x = 2.6 \).