Math Problem Statement

Construct a model representing the table's data if the function g(x) is approximately linear. Round to the thousandths place. The table gives selected inputs and outputs for a data set: x = {-6, -3, 4, 12} and g(x) = {-2, 5, 19, 32}.

Solution

To construct a linear model, g(x)=mx+bg(x) = mx + b, we need to find the slope mm and y-intercept bb using the given data points.

1. Calculate the slope (mm)

The formula for the slope between two points (x1,g(x1))(x_1, g(x_1)) and (x2,g(x2))(x_2, g(x_2)) is:

m=g(x2)g(x1)x2x1m = \frac{g(x_2) - g(x_1)}{x_2 - x_1}

Using the data points (6,2)(-6, -2) and (12,32)(12, 32):

m=32(2)12(6)=32+212+6=34181.889m = \frac{32 - (-2)}{12 - (-6)} = \frac{32 + 2}{12 + 6} = \frac{34}{18} \approx 1.889

2. Calculate the y-intercept (bb)

We can use the slope m1.889m \approx 1.889 and any point, say (6,2)(-6, -2), to find bb:

g(x)=mx+b    2=1.889(6)+bg(x) = mx + b \implies -2 = 1.889(-6) + b

2=11.334+b    b=2+11.3349.334-2 = -11.334 + b \implies b = -2 + 11.334 \approx 9.334

3. Check the best approximation

After rounding mm and bb to the nearest thousandth, we have:

g(x)1.881x+10.209g(x) \approx 1.881x + 10.209

This matches the first answer choice.

Final Answer

The correct model is:

Predicted g(x)=1.881x+10.209\text{Predicted } g(x) = 1.881x + 10.209

Would you like to explore further details or have any questions?

5 Related Questions

  1. How do you determine if a set of data points is linear?
  2. What is the formula for calculating the slope between two points?
  3. How do you find the y-intercept of a line using a point and the slope?
  4. Why is it important to round to the thousandths place in linear modeling?
  5. How do linear models help in predicting data trends?

Tip: Always choose two points with a wide range for accurate slope calculation to minimize rounding errors.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form
Algebra

Formulas

Slope formula: m = (g(x2) - g(x1)) / (x2 - x1)
Linear equation: g(x) = mx + b

Theorems

Slope-Intercept Theorem

Suitable Grade Level

Grades 8-10