A. Quadratic Functions · Quadratic Functions

Finding a Quadratic Model for Data

A3.2 A3.1About 75 minutes

Learning goals

I can find the equation of a quadratic model from a scatter plot using an estimated vertex or estimated zeros.
I can compare my model with a regression equation from technology.
I can judge a model using residuals, R² and the context.
I can use a model to interpolate and extrapolate, and explain the risks.

Words to know

quadratic regression
A technology method that finds the quadratic y = ax² + bx + c that best fits a set of data.

R² (coefficient of determination)
A number from 0 to 1 showing how closely data fits a regression curve; closer to 1 is closer.

residual
Measured value minus predicted value.

interpolation / extrapolation
Predicting inside / outside the range of the data.

curve of best fit
A smooth curve that models the trend of the data.

From dots to a formula

In Collecting and Graphing Data That Can Be Modelled by a Quadratic, you gathered data and saw it curve like a parabola. A graph is useful, but an equation is more powerful: it lets you predict values you did not measure, find the exact maximum, and compare experiments.

In this lesson you will find the equation of the quadratic function that best models a data set, by hand using what you know about the forms, and then compare it with the curve of best fit that technology produces.

Strategy 1: vertex form (when you can see the vertex)

Draw a smooth curve of best fit on the scatter plot.
Estimate the vertex (h, k) from the curve (not necessarily a data point).
Write y = a(x − h)² + k.
Choose another point on the curve (or a reliable data point far from the vertex), substitute it, and solve for a.

Strategy 2: factored form (when you can see the zeros)

Estimate the zeros r and s where the curve meets the horizontal axis.
Write y = a(x − r)(x − s).
Substitute another point (often near the top or bottom) and solve for a.

Strategy 3: technology (quadratic regression)

Desmos, a graphing calculator or a spreadsheet can compute the quadratic regression: the curve y = ax² + bx + c that makes the total squared vertical distances from the points as small as possible. In Desmos, type the data into a table, then type y₁ ~ ax₁² + bx₁ + c. The output includes R², a number from 0 to 1; the closer to 1, the closer the points are to the curve.

Worked example

Example 1: A tossed ball (vertex form)

A ball was tossed upward and filmed. Its height h in metres after t seconds:

t: 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2

h: 1.1, 2.1, 3.0, 3.4, 3.5, 3.0, 2.2

Find a quadratic model by hand.

Plot the points and draw a curve of best fit. The highest points are at t = 0.6 and t = 0.8 (3.4 and 3.5), so the curve peaks between them.
Estimate the vertex: about (0.7, 3.5). So h = a(t − 0.7)² + 3.5.
Use a point far from the vertex, (0, 1.1): 1.1 = a(0 − 0.7)² + 3.5, so 0.49a = −2.4 and a ≈ −4.9.
Model: h = −4.9(t − 0.7)² + 3.5.
Test it at t = 1.2: −4.9(0.25) + 3.5 = 2.275 ≈ 2.3. Measured 2.2. Close. ✓

h ≈ −4.9(t − 0.7)² + 3.5.

The ball data with the hand model (solid) and the technology regression (dashed). The two curves are so close that they almost overlap.

Worked example

Example 2: Comparing with technology

Desmos gives the quadratic regression h = −4.94t² + 6.93t + 1.03 with R² ≈ 0.994 for the ball data. Compare it with the hand model h = −4.9(t − 0.7)² + 3.5.

Write the hand model in standard form: −4.9(t² − 1.4t + 0.49) + 3.5 = −4.9t² + 6.86t + 1.10.
Compare coefficients: a: −4.9 vs −4.94; b: 6.86 vs 6.93; c: 1.10 vs 1.03. Very close.
Compare predictions at t = 0.5: hand −4.9(0.04) + 3.5 = 3.30; regression −1.235 + 3.465 + 1.03 = 3.26. They differ by only 0.04 m.
R² ≈ 0.994 is very close to 1, so a quadratic is a very good model for this data.
Notice that a ≈ −4.9 in both. For an object moving freely near Earth's surface, the t² coefficient is about −4.9 when heights are in metres and time is in seconds (half of the acceleration due to gravity, 9.8 m/s²). The data agrees with physics.

The hand model and the regression agree closely; both have a ≈ −4.9.

Worked example

Example 3: A water jet (factored form)

A garden-hose jet of water was photographed against a grid. The height y (cm) of the jet at horizontal distance x (cm) from the nozzle:

x: 0, 20, 40, 60, 80, 100, 120

y: 0, 22, 37, 41, 36, 23, 1

Find a model in factored form.

The jet starts at height 0 at x = 0 and lands at about x = 120. Zeros ≈ 0 and 120.
y = a(x − 0)(x − 120) = ax(x − 120).
The axis is halfway, at x = 60, and the data point (60, 41) is at the top. 41 = a(60)(60 − 120) = −3600a, so a = −41 ÷ 3600 ≈ −0.0114.
Model: y = −0.0114x(x − 120).
Technology gives y = −0.01125x² + 1.357x − 0.071 (R² ≈ 0.999). Our model in standard form is −0.0114x² + 1.368x. Very close.
The regression has c = −0.071, a tiny negative height at the nozzle. The hand model starts exactly at 0, which matches the situation better there.

y ≈ −0.0114x(x − 120).

The water-jet data with the factored-form model. The zeros at about 0 and 120 and the top point (60, 41) are enough to find the equation.

Judging a model

Residuals: residual = measured value − predicted value. Small residuals, some positive and some negative, mean a good fit.
R² close to 1 means the points lie close to the curve. It does not prove the situation is truly quadratic.
Context: does the model make sense at the ends of the domain (a height of 0 at launch, no negative heights)?
Interpolation (predicting inside the data range) is usually reliable. Extrapolation (predicting beyond it) is riskier, because the situation may change: the ball hits the ground, the bottle empties.

Just for fun

Your hand model and the computer's regression disagree in the second decimal place. Congratulations: you are now arguing with a computer about a ball, which is basically what scientists do all day.

Hands-on activity

Model your own data

You need: the data you collected in the ramp activity of Collecting and Graphing Data That Can Be Modelled by a Quadratic (or the ramp data from that lesson: t = 0, 0.5, 1.0, 1.5, 2.0, 2.5 s and d = 0, 4, 15, 34, 61, 94 cm), graph paper, a calculator, and Desmos if you have a device.

Plot the data and draw a curve of best fit.
The ball starts from rest, so the vertex is (0, 0). Write d = at².
Use a point far from the vertex to find a. Try two different points and compare.
Use your model to predict the distance at a time you did not measure (interpolate), and at a time beyond your data (extrapolate). Which prediction do you trust more? Why?
If you have Desmos, enter the data and find the quadratic regression. Compare a, b, c and R² with your model.

What to look for: with the sample data, (2, 61) gives a = 15.25 and (2.5, 94) gives a = 15.04, so d ≈ 15t². At t = 1.75 s this predicts about 46 cm. Extrapolating to t = 4 s predicts 240 cm, which only works if the ramp is at least 2.4 m long. A regression of the sample data gives a close to 15, with b and c close to 0.

Your turn

A curve of best fit has estimated vertex (2, 9) and passes through (0, 1). Find its equation.

A curve of best fit has zeros 1 and 7 and passes through (3, −8). Find its equation.

Find a model for x = 0, 1, 2, 3, 4, 5, 6 and y = 12, 7, 4, 3, 4, 7, 12.

A kicked ball leaves the ground at t = 0, lands at t = 3.0 s, and its highest measured height is 11 m at t = 1.5 s. Find a model in factored form.

A calculator's quadratic regression for the data in the third question is y = x² − 6x + 12. Is it the same as your model?

What does R² = 0.98 tell you about a quadratic regression?

More practice: warm-up

Which form uses an estimated vertex?

Which form uses estimated zeros?

A parabola has vertex (0, 5) and passes through (1, 3). Find a.

Name a free technology tool that can do quadratic regression.

A measured value is 10 and the model predicts 9.4. What is the residual?

More practice: on your own

An arch was measured: x = 0, 10, 20, 30, 40, 50 m and y = 0, 16, 24, 24, 16, 0 m. Find a model and the maximum height.

For the ramp data, use (2, 61) to find a in d = at². Then use (2.5, 94). Compare.

The draining-bottle data (t = 0, 20, 40, 60, 80, 100 s; h = 20, 15.2, 11.0, 7.4, 4.6, 2.4 cm) has regression h = 0.00083t² − 0.259t + 20.02 with R² ≈ 0.99998. Use it to predict the height at t = 50 s.

Use the ball data points (0, 1.1), (0.6, 3.4) and (1.2, 2.2). The hand model predicts 1.10, 3.45 and 2.28; the regression predicts 1.03, 3.41 and 2.23. Find the residuals for each model. Which fits these three points better?

Explain the difference between interpolation and extrapolation, and why extrapolation is riskier.

Challenge

Give a reason why a hand-made model might be more useful than a regression in some situations.

For the tossed-ball regression h = −4.94t² + 6.93t + 1.03, use the physics model h = −4.9t² + v₀t + h₀ to estimate the ball's initial upward speed v₀ and the release height h₀.

Use the bottle regression h = 0.00083t² − 0.259t + 20.02 to estimate when the water level reaches 0. Then explain why the model should not be used after that time.

Check your understanding

Quiz

To find a in y = a(x − h)² + k after estimating the vertex, you…

A curve of best fit has zeros 0 and 10 and passes through (5, 20). What is its equation?

What does quadratic regression do?

Which R² value shows the closest fit?

Predicting a value beyond the range of the data is called…

For heights in metres and time in seconds, a thrown object's model usually has a t² coefficient close to…

For parents and teachers

Big idea: expectation A3.2 asks students to determine, through investigation using a variety of strategies, the equation of the quadratic function that best models a suitable data set graphed on a scatter plot, and to compare this equation to the equation of a curve of best fit generated with technology.

How to help: if you have a computer, Desmos regression is worth ten minutes together: enter a table, type y₁ ~ ax₁² + bx₁ + c, and compare with the hand model. Ask "Which model makes more sense at the start? At the end?"

Common difficulties:

Using a data point that is very close to the estimated vertex to find a (small errors become big ones). Use a point far away.
Reading R² as "the probability the model is right".
Trusting extrapolations without checking the context.
Sign errors with h and with zeros.