Question

(1, 3), (2, 6), (3, 9), and (4, 12) the function modeled by the pattern is_____, since the slope is_______and the y-intercept is_____

Answer 1

To determine the function modeled by the given points \((1, 3)\), \((2, 6)\), \((3, 9)\), and \((4, 12)\), we first notice that these points suggest a linear relationship.

We can represent a linear function in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

1. **Finding the Slope (m)**:
The slope can be calculated from any two points using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((1, 3)\) and \((2, 6)\):
\[
m = \frac{6 - 3}{2 - 1} = \frac{3}{1} = 3
\]

This calculation holds for any pair of points. For example, using points \((2, 6)\) and \((3, 9)\):
\[
m = \frac{9 - 6}{3 - 2} = \frac{3}{1} = 3
\]

So, the slope is \(3\).

2. **Finding the Y-Intercept (b)**:
To find the y-intercept, we can choose one of the points and use the slope to solve for \(b\). Using the point \((1, 3)\):
\[
y = mx + b \implies 3 = 3(1) + b
\]
\[
3 = 3 + b \implies b = 3 - 3 = 0
\]

Thus, the y-intercept is \(0\).

3. **Writing the Function**:
The equation of the line based on the slope and y-intercept found is:
\[
y = 3x + 0 \quad \text{or simply} \quad y = 3x
\]

**Final Answer**:
The function modeled by the pattern is \(y = 3x\), since the slope is \(3\) and the y-intercept is \(0\).