Question

be able to draw valid conclusions, and make inferences from experimental data.
practice: graph the three points given in the
table onto the graph paper given. draw a
straight line between the
three points. identify the missing data.
average radius of trees in a forest

age of trees (years)	average radius (cm)
2	2.6
3
4
5
6	4.4

1. on the above graph, what is the independent

variable?

1. on the above graph, what is the dependent variable?
2. from your graph, predict what the radius of a tree would be if it were 4 years old.
3. what conclusion can you draw from this graph? (in other words, how are the two variables related?)

average radius of trees in a for
(graph: x - axis: age of trees (years) from 0 to 6; y - axis: average radius (cm) from 0 to 5. points plotted and a line drawn.)

Explanation:

Question 10

In a graph, the independent variable is the one that is manipulated or does not depend on other variables. Here, the x - axis represents the age of trees (years), and it is the variable that we can consider as the input or the variable that is not dependent on the radius. So the independent variable is the age of trees.

The dependent variable is the one that depends on the independent variable. In this graph, the y - axis represents the average radius (cm) of the trees, and the radius depends on the age of the tree. So the dependent variable is the average radius of trees.

Step 1: Analyze the pattern

First, we find the slope between the given points. For age 1 (x = 1) with radius 2.2 (y = 2.2) and age 2 (x = 2) with radius 2.6 (y = 2.6). The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2.6 - 2.2}{2 - 1}=\frac{0.4}{1}=0.4$.

Step 2: Predict for age 4

We can use the equation of the line. Using the point - slope form $y - y_1=m(x - x_1)$. Let's use the point (1, 2.2). The equation is $y-2.2 = 0.4(x - 1)$. For $x = 4$, $y-2.2=0.4\times(4 - 1)=0.4\times3 = 1.2$. Then $y=2.2 + 1.2=3.4$? Wait, no, looking at the graph, the line seems to have a slope. Wait, the first point at x = 0, y≈1.8? Wait, maybe better to look at the graph. From the table, at x = 1, y = 2.2; x = 2, y = 2.6. The difference in y is 0.4 per year. So at x = 3, y=2.6 + 0.4 = 3.0? Wait, the graph has a point at x = 2, y≈3.2? Wait, maybe my initial slope calculation is wrong. Wait, looking at the graph, the x - axis is age (years) and y - axis is radius (cm). Let's list the given points: (1, 2.2), (2, 2.6), (6, 4.4). Let's calculate the slope between (1, 2.2) and (6, 4.4). Slope $m=\frac{4.4 - 2.2}{6 - 1}=\frac{2.2}{5}=0.44$. But maybe the pattern is that each year the radius increases by 0.4 cm? Wait, from x = 1 (2.2) to x = 2 (2.6), increase by 0.4. x = 2 to x = 3, increase by 0.4 (2.6+0.4 = 3.0), x = 3 to x = 4, increase by 0.4 (3.0 + 0.4=3.4)? But looking at the graph, the line at x = 4, the y - value seems to be around 3.6? Wait, maybe the correct way is to look at the graph. The graph shows a linear relationship. At x = 1, y≈2.4? Wait, maybe the table has a typo? Wait, the table says at age 1, radius 2.2; age 2, radius 2.6. So the rate of change is (2.6 - 2.2)/(2 - 1)=0.4 per year. So for age 4, starting from age 2 (2.6), after 2 more years (4 - 2 = 2), the radius increases by 0.42 = 0.8. So 2.6+0.8 = 3.4? Or from age 1 (2.2), after 3 years (4 - 1 = 3), 2.2+0.43 = 2.2 + 1.2 = 3.4. Wait, but the graph has a point at x = 4, the y - coordinate is around 3.6? Wait, maybe I made a mistake. Alternatively, looking at the graph, the line passes through (0, 1.8), (1, 2.2), (2, 2.6), (3, 3.0), (4, 3.4), (5, 3.8), (6, 4.2)? But the table says at x = 6, y = 4.4. Anyway, following the arithmetic sequence with common difference 0.4. So at x = 4, y=2.2+(4 - 1)0.4=2.2 + 1.2 = 3.4? Or 2.6+(4 - 2)0.4=2.6 + 0.8 = 3.4. So the predicted radius for a 4 - year - old tree is 3.4 cm? Wait, but maybe the answer is 3.6? Wait, no, let's check the graph. The graph's line at x = 4, the y - value is around 3.6? Wait, maybe my slope is wrong. Let's use the two points (1, 2.2) and (6, 4.4). The equation of the line is $y=mx + b$. Substitute (1, 2.2): $2.2=m + b$. Substitute (6, 4.4): $4.4 = 6m + b$. Subtract the first equation from the second: $4.4-2.2=6m + b-(m + b)\Rightarrow2.2 = 5m\Rightarrow m = 0.44$. Then $b=2.2 - 0.44=1.76$. So the equation is $y = 0.44x+1.76$. For $x = 4$, $y=0.44\times4 + 1.76=1.76+1.76 = 3.52\approx3.5$? But maybe the answer is 3.6 or 4.0? Wait, looking at the graph, the line at x = 4, the y - coordinate is around 3.6? Wait, maybe the intended answer is 3.6? Wait, no, let's see the table. At x = 1, 2.2; x = 2, 2.6; so the difference is 0.4. So x = 3: 2.6+0.4 = 3.0; x = 4: 3.0+0.4 = 3.4; x = 5: 3.4+0.4 = 3.8; x = 6: 3.8+0.6 = 4.4 (which matches the table). Ah, so at x = 6, there is an extra 0.6. But maybe the first point at x = 0, y = 1.8. So the pattern from x = 1 to x = 5 is + 0.4 each year. So at x = 4, y = 3.4.

Answer:

Age of Trees (years)

Question 11