To determine the function rule for the given table, we need to establish the relationship between the temperature (x) and the number of lemonades sold (y). We are dealing with pairs of (temperature, lemonade sold) as follows:
[tex]\[
(55, 1), (65, 11), (70, 21), (75, 31)
\][/tex]
We assume the data can be represented linearly in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
### Step 1: Find the slope [tex]\( m \)[/tex]
The slope [tex]\( m \)[/tex] can be found using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Using the first and last points from the table:
[tex]\[
(55, 1) \quad \text{and} \quad (75, 31)
\][/tex]
[tex]\[
m = \frac{31 - 1}{75 - 55} = \frac{30}{20} = 1.5
\][/tex]
### Step 2: Find the y-intercept [tex]\( b \)[/tex]
To find the y-intercept [tex]\( b \)[/tex], we use the equation of the line [tex]\( y = mx + b \)[/tex] and one of the points. Let's use the point [tex]\((55, 1)\)[/tex]:
[tex]\[
1 = 1.5(55) + b
\][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[
1 = 82.5 + b
\][/tex]
[tex]\[
b = 1 - 82.5 = -81.5
\][/tex]
### Step 3: Write the function rule
With the slope [tex]\( m = 1.5 \)[/tex] and y-intercept [tex]\( b = -81.5 \)[/tex], the function rule is:
[tex]\[
y = 1.5x - 81.5
\][/tex]
### Final Function Rule
So the function that describes the relationship between temperature and the number of lemonade sold is:
[tex]\[
y = 1.5x - 81.5
\][/tex]
