Tree A is curved which means it could be an exponential function. The yy-coordinates are 10, 20, 40, and 80. Since the yy-coordinates are doubling every 10 years, then the function is exponential since exponential functions have a constant factor (the number you multiply by).

2. Unlike the yy-coordinates for Tree A, the yy-coordinates for Tree B are not increasing by a constant factor. Since 37−12=25, 62−37=25, and 87−62=25, then the yy-coordinates for Tree B are increasing by a constant amount of 25. This means the function has a constant rate of change of 25 feet per 10 years. The function is then linear since linear functions have constant rates of change.

Exponential functions are of the form y=a⋅b^(x/c) where aa is the amount when x=0, b is the growth factor, and c is how often the amount changes by the growth factor. Since the y-coordinate is 10 when x=0 (the year 1960), then a=10. Since the y-coordinates are doubling every 10 years, then b=2 and c=10. The function for Tree A is then y=10⋅2^(x/10).

Linear functions are of the form y=mx+b where mm is the constant rate of change and bb is the y-intercept. Since the rate of change is 25 feet per 10 years, then m=25/10=2.5m. Since the height is 12 feet when x=0 (the year 1960), then b=12. The function for Tree B is then y=2.5x+12.

From the previous problems, Tree A is growing by a constant factor of 2 every 10 years and Tree B is growing by a constant amount of 25 feet every 10 years. In the long term, Tree A has a greater growth rate than Tree B since growing by a constant factor will give greater increases than growing by a constant amount as time increases.

5. From the graph, the initial height of Tree A was 10 feet in 1960. From the table, the initial height of Tree B was 12 feet in 1960. Therefore, Tree B had a greater initial height.

6. From the graph and table, in 1990 Tree A had a height of 80 feet and Tree B had a height of 87 feet. Tree A's height will then exceed Tree B's height sometime after 1990. Since 1990 is 30 years after 1960, then x=30x=30 corresponds to 1990. Make a table finding the heights of the trees after 1990:

The height of Tree A will then exceed the height of Tree B after x=33x=33 years which corresponds to 1993 1993 .

2. Unlike the yy-coordinates for Tree A, the yy-coordinates for Tree B are not increasing by a constant factor. Since 37−12=25, 62−37=25, and 87−62=25, then the yy-coordinates for Tree B are increasing by a constant amount of 25. This means the function has a constant rate of change of 25 feet per 10 years. The function is then linear since linear functions have constant rates of change.

Exponential functions are of the form y=a⋅b^(x/c) where aa is the amount when x=0, b is the growth factor, and c is how often the amount changes by the growth factor. Since the y-coordinate is 10 when x=0 (the year 1960), then a=10. Since the y-coordinates are doubling every 10 years, then b=2 and c=10. The function for Tree A is then y=10⋅2^(x/10).

Linear functions are of the form y=mx+b where mm is the constant rate of change and bb is the y-intercept. Since the rate of change is 25 feet per 10 years, then m=25/10=2.5m. Since the height is 12 feet when x=0 (the year 1960), then b=12. The function for Tree B is then y=2.5x+12.

From the previous problems, Tree A is growing by a constant factor of 2 every 10 years and Tree B is growing by a constant amount of 25 feet every 10 years. In the long term, Tree A has a greater growth rate than Tree B since growing by a constant factor will give greater increases than growing by a constant amount as time increases.

5. From the graph, the initial height of Tree A was 10 feet in 1960. From the table, the initial height of Tree B was 12 feet in 1960. Therefore, Tree B had a greater initial height.

6. From the graph and table, in 1990 Tree A had a height of 80 feet and Tree B had a height of 87 feet. Tree A's height will then exceed Tree B's height sometime after 1990. Since 1990 is 30 years after 1960, then x=30x=30 corresponds to 1990. Make a table finding the heights of the trees after 1990:

The height of Tree A will then exceed the height of Tree B after x=33x=33 years which corresponds to 1993 1993 .