Question

A set of points is shown on the graph.

Scatter plot with a point at negative 5 comma negative 1, a point at negative 2 comma 4, a point at 1 comma 9, a point at 3 comma 5, and a point at 4 comma 7.

Which of the following equations is the best model for a line of fit for the data?

y equals 1 fourth times x plus 5.
y equals negative 1 fourth times x plus 5.
y equals 3 fourths times x plus.
y equals negative 3 fourths times x plus 5.

Answer 1

To determine which equation is the best model for a line of fit for the given set of points, we can analyze the coordinates of the points you're provided:

1. \((-5, -1)\)
2. \((-2, 4)\)
3. \((1, 9)\)
4. \((3, 5)\)
5. \((4, 7)\)

From the y-coordinates, we can see that the values seem to increase as the x-values increase, suggesting a positive slope. Let's analyze the provided equations to identify which one fits the data points best.

1. **Equation 1:** \(y = \frac{1}{4}x + 5\) (Positive slope)
2. **Equation 2:** \(y = -\frac{1}{4}x + 5\) (Negative slope)
3. **Equation 3:** \(y = \frac{3}{4}x + ?\) (Positive slope, but incomplete)
4. **Equation 4:** \(y = -\frac{3}{4}x + 5\) (Negative slope)

Since the slope is positive for the first and third equations, we can disregard the second and fourth equations.

Now let's analyze the slopes of the two equations with positive slopes. For better accuracy, let’s estimate a linear fit using the points provided.

To find a rough line of fit, we can identify approximate values of slope and y-intercept based on the coordinates.

Calculating the slope using some points:
- Choosing the points \((-5, -1)\) and \((4, 7)\):

\[
\text{slope} = \frac{7 - (-1)}{4 - (-5)} = \frac{8}{9} \quad \text{(This is approximately } 0.89\text{)}
\]

This slope is not perfectly fitting the slopes of the equations given but indicates that we would expect a slope less steep than \(1\).

Comparing the slopes of the available choices:
- \( \frac{1}{4} = 0.25\)
- \( \frac{3}{4} = 0.75\)

Since none of the slopes calculated are closer to exactly fitting any of the equations of the fit that were provided, we might keep \( y = \frac{3}{4}x + b \) in mind since it’s a positive slope.

Finalizing this analysis (as the challenge doesn’t provide exact completion for the third equation), and noticing that \( y = -\frac{1}{4}x + 5 \) might seem off due to its negative slope.

Overall, without calculations providing exact fits, \(y = \frac{3}{4}x + b\) may fit the cluster closest due to having the minor gradient in regards to the increase present in points.

Thus, if we had to choose, we can conclude:

The best approximate model would likely be \(y = \frac{3}{4}x + b\).

However, based on options and reasoning about the scatter, it is prudent to select **\( y = \frac{3}{4}x + b \)**, given the profound consensus for positive trends.