Understanding and applying statistical models (linear, quadratic, and exponential)

Understanding and applying statistical models (linear, quadratic, and exponential) exercise

• Construct the corresponding function.

  Hints

  If you put in an $x$-value into a function, the result will be its corresponding $y$-value.

  If the change in $y$ over the change in $x$ is always the same, then the resulting function is linear.

  Solution

  Let's take a look at the table recording Phillipe's production process.

  A table represents a linear function if the change in any two consecutive $y$ values divided by the change in their corresponding $x$ values is equal to the same constant value, $m$.

  The change in $y$ is often called the first difference.

  From the table, we can see that the change in any two consecutive $y$ values is always $0.7$, and that the change in their corresponding $x$ values is $1$.

  So in this case, $m$ equals $0.7$ divided by $1$, which is just $0.7$.

  With this information, we can see that the equation representing Phillipe's production process is linear.

  In general, a linear function is given by an equation of the form $y=mx+b$.

  Plugging in $0.7$ for $m$ in the equation, we have the function $y=0.7x+1$, which describes Phillipe's watch production process.

• Determine the ratio of the $y$ values.

  Hints

  Calculate the first two second differences.

  A table represents a quadratic function if the second difference is constant.

  Solution

  First, let's check if this table represents a linear function. It represents a linear function if the change in $y$ over the change in $x$ is constant.

  Since the difference of the $x$ values is $1$, we just need to check to see if the first difference is constant:

  • $1.40-1.00=0.40$
  • $1.96-1.40=0.56$
  • $2.74-1.96=0.78$

  We can already see that the first difference is not constant, so this table does not represent a linear function.

  If a table represents a quadratic function, then the difference of the $x$ values is $1$ and the second difference is constant. Calculating the second difference:

  • $0.56-0.40=0.16$
  • $0.78-0.56=0.22$

  We can already see that the second difference is not constant, so this table does not represent a quadratic function.

  This table represents an exponential function if the difference of the $x$ values is $1$ and the ratio of the $y$ values is constant. Calculating the ratio of the $y$ values, we can see that it is always $1.4$. Thus, this table represents an exponential function.

• Plot the points from the graph and determine if it is linear, quadratic, or exponential.

  Hints

  First, calculate the points when $x=0$, $x=1$, $x=2$, and so on.

  To plot a point $(x,y)$ in a coordinate plane proceed as follows:

  Draw a line parallel to the $y$-axis passing through the $x$-value, and a line parallel to the $x$-axis passing through the $y$-value. The desired point is the intersection of those two lines.

  Connect those points to get the desired graph. The graph of a quadratic function is a parabola.

  Solution

  We are given the following table of points:

  $\begin{array}{c|c|c|c|c} x&0&1&2&3\\ \hline y&0.6&2.1&4.6&8.1 \end{array}$

  To plot each ordered pair $(x,y)$, draw a line parallel to the $y$-axis passing through the $x$-value, and a line parallel to the $x$-axis passing through the $y$-value. The desired point is the intersection of those two lines.

  Lastly, we connect those points to get the resulting graph, which is the graph of a quadratic function.

• Determine the corresponding function.

  Hints

  If the change in $x$ is 1, and the first difference is constant, then the table represents a linear function.

  If the change in $x$ is 1, and the second difference is constant, then the table represents a quadratic function.

  If the change in $x$ is 1, and the ratio of the $y$ values is constant, then the table represents a linear function.

  Solution

  We can see that the differences in the $x$ values are all $1$.

  Checking the first difference, we can see already from the first two,

  • $3.8-2=1.8$, and
  • $7.2-3.8=3.4$,

  that the first difference is not constant.

  Perhaps it's a quadratic function. For this we have to check the second difference. We have the following first differences:

  • $3.8-2=1.8$
  • $7.2-3.8=3.4$
  • $13.7-7.2=6.5$

  From these, we get the following second differences:

  • $3.4-1.8=1.6$
  • $6.5-3.4=3.1$

  We can see already that the second difference is not constant, so this function is not a quadratic function.

  Let's check to see if the table represents an exponential function. Since we already know that the difference of the $x$ values is 1, we just have to check that the ratio of the $y$ values is constant:

  • $\frac{3.8}{2}=1.9$
  • $\frac{7.2}{3.8}\approx1.9$
  • $\frac{13.7}{7.2}\approx1.9$

  Indeed, the ratio of the $y$ values is always $1.9$. So this table represents an exponential function!

• Explain how to distinguish the several functions.

  Hints

  Here you see a table for a quadratic function.

  The second difference of a table representing a quadratic function is constant.

  Solution

  Let's go through each statement one by one:

  • For a linear function, the change in $y$ is constant. This statement is false. For a linear function, the change in $y$ over the change in $x$ is constant. This subtle difference is important to remember.

  • A table represents a linear function if the change in any two consecutive $y$ values divided by the change in their corresponding $x$ values is constant. This statement is true.

  • A table represents a quadratic function if the the change in $x$ is constant and the change in the first difference is also constant. This statement is true.

  • For quadratic functions, the change in $y$ is constant. This statement is false. For quadratic functions, the change in the change in $y$, or second difference, is constant.

  • The table represents an exponential function if the change in $x$ is always constant and the ratio of the $y$ values is constant. This statement is true.

  • For exponential functions, the change in the first difference is constant. This statement is false. For exponential functions, the ratio of the $y$ values is constant.

• Determine if the table represents a linear, quadratic, or exponential function.

  Hints

  If the change in $y$ over the change in $x$ is constant, then the function is linear.

  There is one linear function, two quadratic functions, and one exponential function.

  Solution

  Let's start with the first table. We can see that the change in $y$ over the change in $x$ is always $3$. And so this table represents a linear function.

  The change in $y$ over the change in $x$ of the second table is not constant:

  • $\frac{49-28}{1}=21$
  • $\frac{28-13}{1}=15$
  • $\frac{13-4}{1}=9$
  • $\frac{4-1}{1}=3$

  If we look at the second difference, we see that it is always equal to $6$:

  • $21-15=6$
  • $15-9=6$
  • $9-3=6$

  Thus, as we also know that the difference of the $x$ values is $1$, this table represents a quadratic function.

  The third table represents an exponential function. How can we confirm this? We can see that the difference in the $x$ values is $1$, and that the ratio of the $y$ values is constant:

  • $\frac{81}{27}=3$
  • $\frac{27}9=3$
  • $\frac93=3$
  • $\frac31=3$

  The fourth table shows a quadratic function once again. Noting that the difference in the $x$ values is always $1$, to see that it is indeed quadratic, we first calculate the first difference:

  • $29-19=10$
  • $19-11=8$
  • $11-5=6$
  • $5-1=4$

  And then calculate the second difference,

  • $10-8=2$
  • $8-6=2$
  • $6-4=2$

  to see that the second difference is constant.