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5. Exponential & Piecewise Functions
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United States of AmericaTennessee
Integrated Math 2

5.06 Scatter plots and fitted functions

Lesson
Practice
Lesson

Concept summary

Regression analysis is used to study the relationship between paired quantities, usually represented in the form \left(x,y\right). The x-variable is the independent variable and the y-variable is the dependent variable. This data can be graphed in a scatter plot and an equation, called the regression model, can be found that best fits the data.

Regression model

Describes the relationship between values of x and y, from which the most probable value of y can be predicted for any value of x.

We can use technology to find nonlinear regression models to fit a given set a data. For now, our nonlinear models will include polynomial (quadratic or cubic), exponential, or radical models.

Coefficient of determination {(R\text{-squared}})

A statistical measure of how close the data values are to the fitted regression model. This value represents the percentage of the variation in y that can be explained by the variation in x.

The value of R^2 can be anything from 0 to 100\%. The closer R^2 is to 1, the better fit the regression model is to the data. For a linear regression model, R^2 is the square of the correlation coefficient, r.

Worked examples

Example 1

Carlos is trying to determine the optimum angle he should kick a soccer ball from out of his hands to achieve the maximum distance. He records 10 kicks and analyses them to determine the angle of trajectory and also the distance travelled. His results are recorded in the table below:

Angle (degrees)24303337434851566064
Distance (feet)112129138155161164158148134124
a

What type of function would best model the data?

Approach

We can determine if the data suggests a quadratic association by plotting the points on a coordinate plane and determining if the data resembles a parabola. We can also use technology to calculate the regression with a polynomial of degree 2. If the value of R^2 is close to 1 then there is likely to be a quadratic association.

Solution

25
30
35
40
45
50
55
60
65
70
x
60
70
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100
110
120
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160
170
y

Yes, a quadratic model is appropriate because when drawn on a graph it has a parabolic shape which is symmetric.

Also, if we calculate regression on this data set with a polynomial of degree 2, we get R^2=0.9662 which is very close to 1, so there is a strong correlation to the quadratic model suggesting a quadratic association.

b

Using technology, determine an appropriate equation to model the data set to four decimal places.

Solution

y=-0.1132x^2+10.3245x-74.5885, where x is the angle in degrees and y is the distance travelled.

c

Calculate and interpret the meaning of the vertex of the model.

Approach

The vertex is the maximum height of the parabola and occurs on the line of symmetry, when x=-\dfrac{b}{2a}. We can then substitute this value into the equation found in part (b) to find y, the maximum distance.

Solution

The vertex represents the optimum angle to kick the ball to achieve the maximum distance travelled.

\displaystyle x\displaystyle =\displaystyle \frac{b}{2a}Line of symmetry
\displaystyle x\displaystyle =\displaystyle -\frac{10.3245}{2\left(-0.1132\right)}Substitute a=-0.1132, b=10.3245
\displaystyle x\displaystyle \approx\displaystyle 45.6Simplify

We can now substitute x=45.6 into the equation to solve for y.

\displaystyle y\displaystyle =\displaystyle -0.1132x^2+10.3245x-74.5885State the equation
\displaystyle y\displaystyle =\displaystyle -0.1132\left(45.6\right)^2+10.3245\left(45.6\right)-74.5885Substitute x=45.6
\displaystyle y\displaystyle \approx\displaystyle 160.8Simplify

The vertex is a maximum and occurs at about \left(45.6, 160.8\right) which means that the maximum distance of 160.8 feet is achieved by kicking the ball at an angle of 45.6 \degree.

Outcomes

M2.N.Q.A.1

Use units as a way to understand real-world problems.*

M2.N.Q.A.1.D

Choose an appropriate level of accuracy when reporting quantities.

M2.S.ID.A.1

Represent data from two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.*

M2.MP2

Reason abstractly and quantitatively.

M2.MP4

Model with mathematics.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

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