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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 8

Quadratic Regression

Use technology to fit a quadratic, interpret a, b, and c, and describe strengths and weaknesses of the model.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Quadratic regression

A calculator can fit y = ax² + bx + c to a list of points. You type the data, choose quadratic regression, and read a, b, and c.

When it is reasonable

Second differences nearly constant, or a scatterplot that looks like a parabola, not a line or an exponential curve.

Strengths

The model can interpolate between data points, name a vertex (max or min), and summarize a curved trend with one equation.

Weaknesses

It can miss points, fail outside the data (extrapolation), and force a parabola onto a pattern that is not quadratic.

Level 1

Easy

Read a, b, c from a reported fit, or spot a quadratic table.

Example
A calculator reports y = 1.02x² − 3.96x + 5.01 for a data set. About what quadratic were the points following, and what is the y-intercept of the model?

Near y = x² − 4x + 5

When x = 0, y ≈ 5.01, so the y-intercept is about 5.

Answer: ≈ x² − 4x + 5; intercept ≈ 5

1
The table is x: 0, 1, 2, 3 and y: 5, 8, 13, 20. First differences 3, 5, 7. Is a quadratic fit reasonable?
Show Answer

Second differences 2, 2. Yes, constant second differences.

Answer: yes

2
A fit is y = −2x² + 12x + 3. Does the model open up or down, and is the vertex a max or a min?
Show Answer

a = −2 < 0, opens down, maximum.

Answer: down; maximum

3
Points (0, 4), (1, 6), (2, 8), (3, 10). Quadratic regression or linear?
Show Answer

First differences 2, 2, 2. Linear. A quadratic fit is unnecessary.

Answer: linear

4
A reported model is y = 0.5x² + 1. Name y when x = 0 and when x = 2.
Show Answer

y(0) = 1

y(2) = 0.5(4) + 1 = 3

Answer: 1 and 3

5
Why might y = 2x² + 3x + 1 fit (0, 1), (1, 6), (2, 15) exactly?
Show Answer

Check: x = 0 gives 1. x = 1 gives 2 + 3 + 1 = 6. x = 2 gives 8 + 6 + 1 = 15.

Three points that already lie on that parabola.

Answer: the points lie on it

Level 2

Medium

Interpret a, the vertex, or a residual in context.

Example
A ball’s height data (t seconds, h feet) is fit by h = −16t² + 40t + 5. What do −16, 5, and the vertex time mean?

−16 is the stretch from gravity in this units system (opens down).

5 is the starting height when t = 0.

t = −402(−16) = 54

The model’s maximum is at 54 seconds.

Answer: start 5 ft; max at 5/4 s

1
Data: (1, 3), (2, 6), (3, 11), (4, 18). Second differences 2, 2. Write the exact quadratic y = x² + 2.
Show Answer

Check: 1 + 2 = 3, 4 + 2 = 6, 9 + 2 = 11, 16 + 2 = 18.

A regression on exact quadratic data should return a = 1, b = 0, c = 2.

Answer: y = x² + 2

2
A calculator fit is y = 1.1x² − 0.2x + 4.8. A point (2, 8) was in the list. Find the residual actual − predicted.
Show Answer

Predicted: 1.1(4) − 0.2(2) + 4.8 = 4.4 − 0.4 + 4.8 = 8.8

Residual 8 − 8.8 = −0.8

Answer: −0.8

3
A garden-area experiment lists width x and area A. The fit is A = −1x² + 24x. What width maximizes the model, and what is a weakness if x must be a whole number of feet?
Show Answer

x = −242(−1) = 12

A(12) = 144

The model is continuous. If only whole-number widths are possible, check x = 11, 12, 13 in the data, not only the vertex.

Answer: 12 ft; discrete widths

4
Why is quadratic regression a poor idea for (0, 2), (1, 4), (2, 8), (3, 16)?
Show Answer

Constant ratio 2. Exponential, not quadratic. Second differences would grow, not stay constant.

Answer: exponential, not quadratic

5
A fit y = 0.8x² + 2.1x + 0.4 is used to predict y at x = 20, far past the data 0 ≤ x ≤ 5. Name the risk.
Show Answer

Extrapolation. The real process may not stay quadratic that far out.

Answer: extrapolation

Level 3

Hard

Compare models, or describe strengths and weaknesses in a story.

Example
Points (0, 6), (1, 9), (2, 14), (3, 21) fit y = x² + 2x + 6 exactly. A linear regression on the same points is worse. Explain using differences, then name one strength of the quadratic model.

First differences 3, 5, 7 (not constant).

Second differences 2, 2 (quadratic).

Strength: the quadratic hits every point here, so interpolation between 0 and 3 is exact on this set.

Answer: quadratic hits the points

1
A calculator gives y = −4.9x² + 20x + 1.2 for a toss, in meters. The data only go to x = 3 s. Using the model, the height at x = 8 s is negative. Interpret.
Show Answer

The formula continues below ground.

In the story, height stops at 0 when the object lands.

The model is not valid after the positive zero. That is a weakness of using the quadratic outside the flight.

Answer: not valid after landing

2
Three models are offered for a scatterplot that peaks then falls: linear, exponential, quadratic. Which is the best family, and what should you still check?
Show Answer

A peak then a fall matches a parabola that opens down.

Still check residuals: no leftover curve, and no need for a piecewise story (rise then a different fall).

Answer: quadratic; check residuals

3
Data are (0, 10), (2, 18), (4, 18), (6, 10). Why is a quadratic a natural fit, and what vertex should regression be near?
Show Answer

Symmetric around x = 3, same y at 0 and 6, 2 and 4.

Vertex near (3, something above 18).

Check: a down-opening parabola through those points has axis x = 3.

Answer: vertex near x = 3

4
A quadratic fit has a = 0.02, very small. The scatterplot on 0 ≤ x ≤ 4 looks almost linear. Strength or weakness?
Show Answer

On a short window a small a looks flat, so a line might be simpler and just as useful.

Weakness: the extra parameter may be fitting noise. Strength: if theory says a gentle curve, a = 0.02 can still be the right family.

Answer: maybe overfit on a short window

5
You have four points and a quadratic has three coefficients. Can regression miss a point? Can it hit all four?
Show Answer

Three coefficients cannot hit four general points.

Regression finds a closest parabola. Residuals need not be zero.

If the four happen to lie on a parabola, it can hit all four.

Answer: usually misses at least one

Super Challenge

Super Challenge

1
A student always trusts the calculator’s quadratic regression because R² is close to 1. Give two reasons that can still be a weak model.
Show Answer

R² can be high on a tiny data set even if the family is wrong.

The fit can fail outside the window, or a simpler line could be close enough.

A leftover pattern in residuals (a curve still in the errors) means the model is missing structure.

Answer: window, residuals, simpler model

2
Design a three-point data set that a quadratic will fit exactly, and a fourth point that would force a nonzero residual.
Show Answer

(0, 1), (1, 2), (2, 5) lie on y = x² + 1.

Add (3, 9) instead of 10. Residual at x = 3 is 9 − 10 = −1 if you keep y = x² + 1, and a new regression would move a, b, c a little.

Answer: e.g. (0, 1), (1, 2), (2, 5), (3, 9)

3
In a profit story, quadratic regression gives P(n) = −2n² + 40n − 50. Interpret the vertex, then name a weakness if n must be a whole number of items and n ≥ 0.
Show Answer

n = −402(−2) = 10

P(10) = 150

The model allows n = 10.5, which is not a number of items. Check n = 9, 10, 11 in the original data.

Answer: max near 10 items; n is discrete

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