Question

the graph below shows a relationship between x and y. which function is the best fit for the data? y = -\frac{1}{4}x + 3 y = -\frac{1}{2}x + 4 y = x + 3 y = -\frac{2}{3}x + 5

Explanation:

Step1: Observe data trend

The data points show a decreasing trend.

Step2: Analyze function slopes

Check the slope of each given linear function. Positive - slope functions are increasing and negative - slope functions are decreasing.

Step3: Select best - fit

Choose the function with a negative slope that best matches the data trend. In this case, \(y=-\frac{1}{2}x + 4\) is the best fit.

Answer:

We need to analyze the trend of the data - points on the graph to determine the best - fit function.

1. First, observe the general trend of the data points:

• The data points seem to follow a linear - like decreasing trend.

1. Now, test each function:

• For \(y = \frac{2}{3}x+5\), it is a linear function with a positive slope (\(m=\frac{2}{3}\)), so it is an increasing function. Since our data points are decreasing, this is not a good fit.
• For \(y=\frac{1}{4}x + 3\), it is a linear function with a positive slope (\(m = \frac{1}{4}\)), so it is an increasing function. This is not a good fit as our data is decreasing.
• For \(y=-\frac{1}{2}x+4\), it is a linear function with a negative slope (\(m=-\frac{1}{2}\)). As \(x\) increases, \(y\) decreases.
• For \(y=x + 3\), it is a linear function with a positive slope (\(m = 1\)), so it is an increasing function. This is not a good fit for our decreasing data set.
• By observing the data points and the nature of the functions, the function \(y=-\frac{1}{2}x + 4\) is most likely to be the best - fit function for the given data as it has a negative slope which matches the decreasing trend of the data points.