To determine the rule for the function given the set of points [tex]\((x, y)\)[/tex] [tex]\(\{(0, -3), (1, 4), (2, 11), (3, 18)\}\)[/tex], we can assume a linear relationship of the form [tex]\( y = mx + c \)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept. Let's determine these coefficients, [tex]\(m\)[/tex] and [tex]\(c\)[/tex], step by step.
### Step 1: Set Up the Linear Equations
Using the points given, we can set up a series of equations based on the linear form [tex]\( y = mx + c \)[/tex]:
1. For the point [tex]\((0, -3)\)[/tex]:
[tex]\[
-3 = m \cdot 0 + c \implies c = -3
\][/tex]
2. For the point [tex]\((1, 4)\)[/tex]:
[tex]\[
4 = m \cdot 1 + c \implies 4 = m + c
\][/tex]
3. For the point [tex]\((2, 11)\)[/tex]:
[tex]\[
11 = m \cdot 2 + c \implies 11 = 2m + c
\][/tex]
4. For the point [tex]\((3, 18)\)[/tex]:
[tex]\[
18 = m \cdot 3 + c \implies 18 = 3m + c
\][/tex]
### Step 2: Substitute [tex]\( c \)[/tex]
From the equation derived from the point [tex]\((0, -3)\)[/tex], we know [tex]\( c = -3 \)[/tex]. Substitute [tex]\( c \)[/tex] into the other equations:
1. For [tex]\((1, 4)\)[/tex]:
[tex]\[
4 = m + (-3) \implies 4 = m - 3 \implies m = 7
\][/tex]
2. For [tex]\((2, 11)\)[/tex]:
[tex]\[
11 = 2m - 3 \implies 11 = 2(7) - 3 \implies 11 = 14 - 3 \implies 11 = 11 \quad \text{(true)}
\][/tex]
3. For [tex]\((3, 18)\)[/tex]:
[tex]\[
18 = 3m - 3 \implies 18 = 3(7) - 3 \implies 18 = 21 - 3 \implies 18 = 18 \quad \text{(true)}
\][/tex]
### Step 3: Construct the Function Rule
Having determined [tex]\( m = 7 \)[/tex] and [tex]\( c = -3 \)[/tex], we can now construct the function rule. The function that passes through the given points is:
[tex]\[
y = 7x - 3
\][/tex]
Thus, the rule for this function is:
[tex]\[
y = 7x - 3
\][/tex]
