Applications of Slope and Intercept

1. The pizza reached the desired temperature of $80^{\circ}$ degrees Celsius after $12$ minutes. This is the same as saying that when Time was $12$, Temperature was $80$.

Therefore, the graph of the relationship should pass through the point $(12, 80)$.

We are also given the rate of change of the relationship: $7.5^{\circ}$ degrees Celsius per minute. This means that when Time increases by $1$, Temperature increases by $7.57$.

Therefore, the graph should also pass through the point $(12+1,80+7.5)$, which is the point $(13,87.5)$. However, this point doesn't appear on the grid! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time decreases by $2$, Temperature decreases by $2 \cdot 7.5=15$. Therefore, the graph should pass through the point $( 12-2,80-15)$, which is the point $(10,65)$.

Now we can define the graph of the relationship.

2. The water level reached the desired height of $220$ centimeters after $20$ minutes. This is the same as saying that when Time was $20$, Water level was $220$.

Therefore, the graph of the relationship should pass through the point $(20, 220)$.

We are also given the rate of change of the relationship: $6$ centimeters per minute. This means that when Time increases by $1$, Water level increases by $6$.

Therefore, the graph should also pass through the point $(20+1, 220+6)$, which is the point $(21, 226 )$. However, this point doesn't appear on the grid! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time decreases by $5$, Water level decreases by $5 \cdot 6=30$. Therefore, the graph should pass through the point $(20-5, 220-30)$, which is the point $( 15, 190)$.

Now we can define the graph of the relationship.

3. Sean finished all the slushy after $50$ seconds. This is the same as saying that when Time was $50$, Slushy was $0$.

Therefore, the graph of the relationship should pass through the point $(50, 0)$.

We are also given the rate of change of the relationship: $5$ milliliters per second. This means that when Time increases by $1$, Slushy decreases by $5$.

Therefore, the graph should also pass through the point $(50+1, 0-5)$, which is the point $(51, -5)$. However, this point doesn't appear on the grid (and it doesn't make sense either)! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time decreases by $5$, Slushy increases by $5 \cdot 5=25$. Therefore, the graph should pass through the point $(50-5,0+25)$, which is the point $(45, 25)$.

Now we can define the graph of the relationship.

4. When Rip van Winkle fell asleep his beard was $8$ millimeters long. This is the same as saying that when Time was $0$, Beard was $8$.

Therefore, the graph of the relationship should pass through the point $(0,8)$.

We are also given the rate of change of the relationship: $2$ millimeters per week. This means that when Time increases by $1$, Beard increases by $2$.

Therefore, the graph should also pass through the point $(0+1,8+2)$, which is the point $(1,10)$. However, this point doesn't appear on the grid! We need to find an equivalent pair of corresponding changes that would help us find a point that we can graph.

Note, for instance, that when Time increases by $2$, Beard increases by $2 \cdot 2=4$. Therefore, the graph should pass through the point $(0+2,8+4)$, which is the point $(2, 12)$.

Now we can define the graph of the relationship.