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408 Quadratic Functions
Lesson
Fitting a Quadratic
Model to Data
Chapter 6
BIG IDEA Quadratic regression is like linear regression in that it finds the model with the least sum of squares of differences from the given data points to the values predicted by the model.
In Chapter 3, you learned how to find an equation for the line through two points, and how to find a linear model for data that lie approximately on a straight line. You can also fit a quadratic model to data that lie approximately on a parabola. Quadratic regression fits a model of the form y = ax^2 + bx + c to data.
MATERIALS old CD that can get scratched up, two 8 1 __ 2 – inch by 11–inch pieces
of paper, quarter, CAS or graphing calculator You are going to investigate how the radius x of a circular object affects the probability y of the object landing completely in a fixed region when dropped. Step 1 A CD like the one pictured at the right has 3 circles on it. Circle I represents the hole in the CD. Circle M (for middle) represents the start of the silver writing surface. Circle O represents the outer edge of the CD. Draw a line across the middle of your 8 1 __ 2 – by – 11 piece of paper, dividing it into two 5 1 __ 2 – by – 8 1 __ 2 rectangular targets. Step 2 On a separate sheet of paper, make four columns labeled Drop Number, I, M, and O. In the Drop Number column, write the integers 1 through 25. Place your divided sheet of 8__^12 – inch – by – 11 – inch paper on the floor and stand above it holding the CD waist high, parallel to the floor. Drop the CD.
Mental Math
Find the slope of the image of 3x + 2 y = 7 under the transformation. a. T6, – 1 b. R 90 c. R 270 d. r (^) y = x
Mental Math
Find the slope of the image of 3x + 2 y = 7 under the transformation. a. T6, – 1 b. R 90 c. R 270 d. r (^) y = x
ActivityActivity
II MM OO
Vocabulary quadratic regression
Fitting a Quadratic Model to Data 409
Lesson 6– 6
If no part of the CD is touching the paper, drop it again. If any part of the CD is touching the paper, determine the score for each circle, I, M, and O as follows: If a circle on the CD lands completely inside one of the two rectangular targets, give it 1 point for that drop. Pictured below are the four possible situations and scores after a drop. The horizontal line represents either the edge of the paper or the line you drew on the paper.
Drop the CD 25 times and fill in your table.
Step 3 Calculate the relative frequency of a circle landing completely in a target. For example, if M has 16 points, then its relative frequency of points per drop is __^1625. Record the frequencies in a table like the one at the right. The radii of circles on a standard CD have been filled in for you, but you should measure your CD to check.
Step 4 Create a scatterplot with three data points (radius, frequency) for I, M, and O on your calculator. Choose the quadratic regression option from the appropriate menu. A sample is shown at the right. On this calculator, the graph of a quadratic model for the data is added to the scatterplot. The calculator displays the equation for the quadratic model.
Step 5 Examine the scatterplot with the graph of the regression equation on it. How well does your model fit your data?
Step 6 Measure the diameter of a quarter and use your regression equation to predict the relative frequency of the quarter landing inside a target rectangle. Drop the quarter 25 times and see if the relative frequency is close to your prediction. Combine your results with other classmates. Compare the combined data with the prediction. Which value is closer to the predicted value—your own data or the combined data?
I 1
M 1
O 1
I 1
M 1
O 0
I 1
M 0
O 0
I 0
M 0
O 0
I and M in target
I only in target
all circles in target
circle point
no circles in target
I 1
M 1
O 1
I 1
M 1
O 0
I 1
M 0
O 0
I 0
M 0
O 0
I and M in target
I only in target
all circles in target
circle point
no circles in target
Circle Radius(in.) FrequencyRelative
I __ 165? M __^78? O^2?
Circle Radius(in.) FrequencyRelative
I __ 165? M __^78? O^2?
Fitting a Quadratic Model to Data 411
Lesson 6– 6
- Solve the system of the Example to find an equation for the parabola that contains the given points.
In 3 and 4, solve a system of equations to write an equation for the parabola that contains the given points.
- (4, – 6), (–2, 30), (0, 10) 4. (–1, – 3), (5, 81), (2, 12)
In 5 and 6, use quadratic regression to find an equation for the parabola that contains the given points.
- (3, 2), (–1, 5), (8, 7) 6. (– 6, 5), (–1, 10), (3, 4)
APPLYING THE MATHEMATICS
- A quarterback threw a ball from 5 yards behind the line of scrimmage and a height of 6 feet. The ball was 10 feet high as it crossed the line of scrimmage. It was caught 20 yards past the line of scrimmage at a height of 5 feet off the ground. a. Find an equation that gives the height y of the ball (in feet) when it was x yards beyond the line of scrimmage. b. Graph the equation.
- In The Greedy Triangle by Marilyn Burns, an equilateral triangle keeps changing shape into a regular polygon with more and more sides. As the number of sides of a polygon increase, so do the number of diagonals of the polygon. Recall the formula for the number of diagonals in a polygon. If you have forgotten, you can derive it using quadratic regression. a. Count the number of diagonals in each polygon below and complete the table to the right.
b. Use quadratic regression to find a model for the number of diagonals in a polygon. c. Is this model exact or approximate? d. Use the model to determine the number of diagonals in a 50-sided polygon.
Peyton Manning throwing for the Indianapolis Colts
Peyton Manning throwing for the Indianapolis Colts
Shape Numberof Sides Number ofDiagonals
Triangle?? Quadrilateral?? Pentagon?? Hexagon??
Shape Numberof Sides Number ofDiagonals
Triangle?? Quadrilateral?? Pentagon?? Hexagon??
412 Quadratic Functions
Chapter 6
- Shown below is a table of data on the eight major planets in our solar system. For each planet, the period of the orbit is given in Earth years and the average distance from the Sun is in astronomical units (AU). One AU is approximately the mean distance between Earth and the Sun. Each planet’s distance from the Sun is not constant due to its elliptical orbit, so the average distance is given.
Planet (^) From the Sun (AU) d^ =^ Avg. Distance^ P^ = (in Earth years)^ Period of Orbit
Mercury 0.39 0. Venus 0.72 0. Earth 1.00 1. Mars 1.52 1. Jupiter 5.20 11. Saturn 9.58 29. Uranus 19.20 84. Neptune 30.05 164.
a. Create a scatterplot of the data. Use average distance d as the independent variable and period P as the dependent variable. b. Fit a quadratic model to these data. Graph the model on the same axes as your scatterplot. Does the model seem like it fits the data? Why or why not? c. Pluto, a dwarf planet, is an average 39.48 AU from the Sun. It takes Pluto 248.54 Earth years to orbit the Sun. Does Pluto fit your model? d. Another dwarf planet, Sedna, or 2003 VB12, as it was originally designated, orbits the sun at an average distance of 90 AU. Use your model from Part b to predict how long it takes Sedna to orbit the Sun.
REVIEW
In 10–12, rewrite the equation in vertex form. (Lesson 6–5)
- y = x^2 - 14 x + 53
- y = 3 x^2 - 9 x + 9
- y = x^2 - 4 gx + 3 g^2
- Find an equation in standard form for the image of the parabola with equation y = – 2 x^2 under the translation T – 4, – 0.. (Lessons 6–3, 6–1)
Pluto and one of its moons, Charon, are shown above. Sunlight takes about
5 __^12 hours to reach Pluto.
Sunlight reaches Earth in about 8 minutes.
Pluto and one of its moons, Charon, are shown above. Sunlight takes about
5 __^12 hours to reach Pluto.
Sunlight reaches Earth in about 8 minutes.
