Question

the table shows the temperature of a greenhouse where the temperature was taken after x hours.
greenhouse temperature

hour, x	temperature (°f), y
5	60
8	66
9	68
13	76
15	80

when these data are graphed on a coordinate grid, the points all lie on the same line.
complete the statements below about the data by selecting the correct choice from each drop - down menu.

• the initial value, or y - intercept, of the linear function is dropdown and represents the starting dropdown
• the rate of change is dropdown and represents dropdown

Explanation:

Step1: Find the y - intercept

The y - intercept of a linear function \(y = mx + b\) is the value of \(y\) when \(x = 0\). From the table, when \(x = 0\), \(y=50\). So the y - intercept \(b = 50\). This represents the starting temperature of the greenhouse (since \(x = 0\) hours is the start time).

Step2: Calculate the rate of change (slope)

The rate of change (slope \(m\)) of a linear function is given by the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points, say \((x_1,y_1)=(0,50)\) and \((x_2,y_2)=(5,60)\). Then \(m=\frac{60 - 50}{5 - 0}=\frac{10}{5}=2\). We can check with another pair, e.g., \((5,60)\) and \((8,66)\): \(m=\frac{66 - 60}{8 - 5}=\frac{6}{3}=2\). So the rate of change is \(2\), which represents the number of degrees Fahrenheit the temperature increases per hour (since for each increase of 1 in \(x\) (hour), \(y\) (temperature) increases by 2).

Answer:

• The initial value, or y - intercept, of the linear function is \(50\) and represents the starting temperature of the greenhouse.
• The rate of change is \(2\) and represents the number of degrees Fahrenheit the temperature increases per hour.