Answer:
Step-by-step explanation:
To determine which equation most closely matches the data in the table, we need to first analyze the table to understand the relationship between the variables. Then, we can compare this relationship with possible equations.
Assuming we have a table like this (since no specific table was provided, this is a general approach):
xx yy
1 2
2 4
3 8
4 16
Steps to Determine the Best-Fit Equation
Identify the Pattern or Trend in the Data: Check if the yy-values follow a particular mathematical trend such as linear, quadratic, exponential, etc.
Hypothesize Possible Equations:
Linear: y=mx+by=mx+b
Quadratic: y=ax2+bx+cy=ax2+bx+c
Exponential: y=abxy=abx
Power: y=axky=axk
Fit Each Hypothesis to the Data:
Use the method of least squares or regression to find the best-fit parameters for each hypothesis.
Compute the correlation coefficient or residuals to measure how well each equation fits the data.
Compare the Fit: Determine which equation has the best fit by looking at the residuals or correlation coefficient. The smaller the residuals or the closer the correlation coefficient to 1, the better the fit.
Analyzing the Sample Data
Given the table above:
xx yy
1 2
2 4
3 8
4 16
We observe the following:
The yy-values seem to double as xx increases by 1.
This suggests an exponential relationship.
Hypothesis 1: Exponential Model y=abxy=abx
From the table:
At x=1x=1, y=2y=2.
At x=2x=2, y=4y=4.
Using these points, we can set up the following equations:
2=ab12=ab1
4=ab24=ab2
Dividing the second equation by the first:
42=ab2ab24=abab2
2=b2=b
Substitute b=2b=2 into the first equation:
2=a⋅22=a⋅2
a=1a=1
Thus, the exponential equation is:
y=1⋅2xy=1⋅2x
y=2xy=2x
Hypothesis 2: Power Model y=axky=axk
From the table:
At x=1x=1, y=2y=2.
At x=2x=2, y=4y=4.
Using these points, we can set up the following equations:
2=a⋅1k2=a⋅1k
4=a⋅2k4=a⋅2k
From the first equation:
a=2a=2
Substituting a=2a=2 into the second equation:
4=2⋅2k4=2⋅2k
2=2k2=2k
k=1k=1
Thus, the power equation simplifies to:
y=2xy=2x
Hypothesis 3: Linear Model y=mx+by=mx+b
A linear model may not be as appropriate given the observed growth pattern, but we can still check it:
Using the points (1,2)(1,2) and (2,4)(2,4).
The slope mm is:
m=4−22−1=2m=2−14−2=2
Using the point (1,2)(1,2) to find the intercept bb:
2=2(1)+b2=2(1)+b
b=0b=0
Thus, the linear equation is:
y=2xy=2x
Hypothesis 4: Quadratic Model y=ax2+bx+cy=ax2+bx+c
Using three points to fit a quadratic curve:
(1,2)(1,2), (2,4)(2,4), (3,8)(3,8).
Setting up the system of equations:
2=a(1)2+b(1)+c2=a(1)2+b(1)+c
4=a(2)2+b(2)+c4=a(2)2+b(2)+c
8=a(3)2+b(3)+c8=a(3)2+b(3)+c
Solving these simultaneously will give the coefficients aa, bb, and cc.
Comparison of Hypotheses
Exponential Model y=2xy=2x:
Fits the given data points exactly.
Power Model y=2xy=2x:
Fits the data only at x=1x=1 and x=2x=2. At x=3x=3, y=6y=6, which deviates from the table value of 8.
Linear Model y=2xy=2x:
As with the power model, it deviates from the table at x=3x=3 and x=4x=4.
Quadratic Model:
Solving for coefficients aa, bb, and cc will be complex but let's check if it closely matches after solving the system.
Given the exponential pattern observed (doubling of yy as xx increases by 1), the exponential model y=2xy=2x is the most likely candidate that fits the data points exactly.
Therefore, the equation that most closely matches the given data is:
y=2xy=2x