Destiny planted a tree in her backyard. It grows at a constant rate each year. Draw a graph and write an equation to represent this situation.

Question

The table shows the relationship between the number of years and the height of Destiny's tree. After 2 years, the tree is 80 inches. After 8 years, it is 170 inches. Plot the two points on the graph to show the height of the tree after 2 years and 8 years.
Number of years Height (inches)
2 80
8 170
You got it!
Each year, the height of the tree increases at a constant rate. So, we can draw a line through these two points to show the linear relationship between the number of years and the height of the tree. Let’s write an equation to represent this line in the form y=mx+b. Start by finding m, or the slope. What is the slope of this line?
x
y
(2, 80)(8, 170)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Number of years
Height (inches)
y= mx + b

Slope =
Good work!
Substitute the slope for m in our equation, or the rate of change of the height of the tree.
x
y
(2, 80)(8, 170)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Number of years
Height (inches)
y= mx + b
y=
x+b

Slope = 15
Good work!
Now let’s solve for b in our equation, or the vertical intercept. Let’s use the point (2, 80). Start by substituting the value of x and y into the equation.
x
y
(2, 80)(8, 170)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Number of years
Height (inches)
y
=
15x
+
b

80
=
15
·

2
+
b
Excellent!
Now, solve for the value of b, or the vertical intercept.
x
y
(2, 80)(8, 170)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Number of years
Height (inches)
y=15x+b

80
=
15·2+b

b
=
Excellent!
Let’s finish writing our equation to represent the relationship between the number of years and the height of the tree. Now that we know the value of b, or the vertical intercept, substitute the vertical intercept for b in our equation
x
y
(2, 80)(8, 170)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Number of years
Height (inches)
y
=
15x + b
y
=
15x +
b
=
50
Nice!
You just wrote an equation to determine the height of the tree, y, after x years. What does this equation mean?
x
y
(2, 80)(8, 170)
0
1
2
3
4
5
6
7
8
9
10
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
Number of years
Height (inches)
y= mx + b
y= 15x + 50

The tree was ?
inches when Destiny planted it, and it grew
? inches each year.

Answer 1

The equation you derived, which represents the height of Destiny's tree in inches after \( x \) years, is:

\[ y = 15x + 50 \]

Interpretation of the Equation:

1. Initial Height when Planted (y-intercept, b):

   • The value of \( b = 50 \) indicates that when Destiny planted the tree (\( x = 0 \)), the tree was 50 inches tall.

2. Rate of Growth (slope, m):

   • The slope \( m = 15 \) signifies that the tree grows 15 inches each year.

Summary:

• The tree was 50 inches when Destiny planted it.
• It grew 15 inches each year.