Topics
9. Variation
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VCE 11 General 2023

9.08 Data models

Lesson
Worksheet
Practice
Lesson

Data model

When linearising data, it may be necessary to determine the equation for the transformation so other values can be predicted. This often requires being able to determine an equation for these types of functions from a set of points or features from the data.

Examples

Example 1

Consider the data:

x012345
y139193351

Using an appropriate linearising transformation, find the equation of the line for the transformed data.

Worked Solution
Create a strategy

Plot the points from the table of values and draw the curve through each plotted point. Then determine its data transformation to linearise it.

Find the slope and y-intercept using the transformed data.

Apply the idea
1
2
3
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5
6
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8
9
x
5
10
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25
30
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40
45
50
y

We can see that the curve that passes through each of the plotted points is parabolic.

So we will use the square transformation.

x012345
x^201491625
y139193351

Here is the table of values of the transformed.

To determine the equation of the line y=m(x^2) + c, from the table of values, when x^2 = 0, \, y=1. So the y-intercept is c=1. Using the coordinate (4,9) from the table, the slope m can be found:

\displaystyle y\displaystyle =\displaystyle m(x^2) + cWrite the equation
\displaystyle 9\displaystyle =\displaystyle m \times 4 + 1Substitute c=1, \, x=4, \, y=9
\displaystyle 8\displaystyle =\displaystyle 4mSubtract 1 from both sides
\displaystyle m\displaystyle =\displaystyle 2DIvide both sides by 4

So, the equation for the transformed data is y=2 \left(x^2 \right) + 1

Example 2

Consider the equation y = 2x^{2}.

a

Complete the following table of values.

x- 2- 1012
y
Worked Solution
Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=-2:

\displaystyle y\displaystyle =\displaystyle 2x^2Write the equation
\displaystyle =\displaystyle 2(-2)^2Substitute x=-2
\displaystyle =\displaystyle 8Evaluate

Similarly, by substituting the remaining x-values into y=2x^2, we get:

x- 2- 1012
y82028
b

Plot the graph.

Worked Solution
Create a strategy

Plot the points from the table of values and draw the curve through each plotted point.

Apply the idea
-3
-2
-1
1
2
3
x
2
4
6
8
10
12
y

The ordered pairs of points are (-2,8),(-1,2),(0,0),(1,2), and (2,8).

This curve of y=2x^2 must pass through each of the plotted points.

Example 3

Consider the function y = \dfrac{1}{x} which is defined for all real values of x except at 0.

a

Complete the following table of values:

x-2-1-\dfrac{1}{2}-\dfrac{1}{4}\dfrac{1}{4}\dfrac{1}{2}12
y
Worked Solution
Create a strategy

Substitute the values from the table into the equation.

Apply the idea

For x=-2::

\displaystyle y\displaystyle =\displaystyle \dfrac{1}{x}
\displaystyle =\displaystyle \dfrac{1}{-2}Substitute x=-2
\displaystyle =\displaystyle -\dfrac{1}{2}Evaluate

Similarly, by substituting the remaining x-values into y=\dfrac{1}{x}, we get:

x-2-1-\dfrac{1}{2}-\dfrac{1}{4}\dfrac{1}{4}\dfrac{1}{2}12
y-\dfrac{1}{2}-1-2-4421\dfrac{1}{2}
b

Plot the points in the table of values.

Worked Solution
Create a strategy

Plot the points from the completed table in part (a).

Apply the idea
-4
-3
-2
-1
1
2
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x
-4
-3
-2
-1
1
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y

The ordered pairs of points to be plotted on the coordinate plane are (-2,-\dfrac{1}{2}), (-1,-1), (-\dfrac{1}{2}, -2), (-\dfrac{1}{4},-4), (\dfrac{1}{4},4), \\ (\dfrac{1}{2},2), \, (1,1), and (2,\dfrac{1}{2}) which are plotted on the graph.

c

Hence draw the curve.

Worked Solution
Create a strategy

Draw the curve passing through each plotted point.

Apply the idea
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

This curve of y=-\dfrac{1}{x} must pass through each of the plotted points and approaches the horizontal asymptote y=0 and the horizontal asymptote x=0.

Idea summary

When linearising data, we should determine the equation for the transformation so other values can be predicted.

Logarithms and technology

Sometimes it may be necessary to use technology to find the equation of a graph, particularly for logarithms. The following example demonstrates this method.

Examples

Example 4

Consider the function y = 5 \log_{10} x.

a

Solve for the x-coordinate of the x-intercept.

Worked Solution
Create a strategy

Substitute y=0 into the equation and then solve for x.

Apply the idea
\displaystyle y\displaystyle =\displaystyle 5 \log_{10} xWrite the equation
\displaystyle 0\displaystyle =\displaystyle 5 \log_{10} xSubstitute y=0
\displaystyle 0\displaystyle =\displaystyle \log_{10} xDivide both sides by 5
\displaystyle x\displaystyle =\displaystyle 10^{0}Write in exponential form
\displaystyle =\displaystyle 1Evaluate
b

Complete the table of values below for y=5 \log_{10} x.

x\dfrac{1}{10}110100
y
Worked Solution
Create a strategy

Substitute each of the x-values in the table into the equation and evaluate.

Apply the idea

For x=\dfrac{1}{10}=0.10:

\displaystyle y\displaystyle =\displaystyle 5 \log_{10} x
\displaystyle =\displaystyle 5 \log_{10} 0.10Substitute x=0.10
\displaystyle =\displaystyle 5 \times -1Evaluate the logarithm
\displaystyle =\displaystyle -5Evaluate

Similarly, by substituting the remaining x-values into 5 \log_{10} x, we get:

x\dfrac{1}{10}110100
y-50510
c

State the equation of the vertical asymptote.

Worked Solution
Create a strategy

Find the value of x in which the graph of the equation approaches but never reaches.

Apply the idea

In the expression \log{10} x, \, x is the result of raising 10 to some power. So regardless of the power, x will be positive. Then the value of x in which the graph of the 5 \log{10} x approaches but never reaches is 0.

So its vertical asymptote is x=0

d

Sketch the graph of y=5 \log_{10} x.

Worked Solution
Create a strategy

Plot the points in the table of values and draw the curve passing through each plotted point.

Apply the idea
10
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50
60
70
80
90
100
x
-6
-4
-2
2
4
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10
y

From the table of values in part (b) we have some the ordered pairs of points to be plotted on the coordinate plane: (\dfrac{1}{10},-5), \, (1,0), \, (10,5) , and (100,10).

This curve of the equation 5\log_{10} x must pass through each of the plotted points.

Idea summary

Sometimes it may be necessary to use technology to find the equation of a graph, particularly for logarithms.

Outcomes

U2.AoS3.6

use a logarithmic (base 10) scale to represent quantities that range over several orders of magnitude and to solve variation problems

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