Question

1. this table shows a linear relationship between the age of a plant in weeks and its height in centimeters for the first 12 weeks.

age (weeks)	height (centimeters)
3	8
10	15
12	17

select all the true statements.

a. the height of the plant increases at a constant rate of 2 cm every week.

b. the height of the plant increases at a constant rate of 1 cm every week.

c. the height of the plant increases at a constant rate of \\(\frac{1}{2}\\) cm every week.

d. age is a function of height.

e. height is a function of age.

Explanation:

Step1: Calculate the rate of change between two points

To find the rate of change (slope) of height with respect to age, we use the formula for slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take the first two points \((1,6)\) and \((3,8)\). Here, \(x_1 = 1,y_1 = 6,x_2=3,y_2 = 8\).
\(m=\frac{8 - 6}{3 - 1}=\frac{2}{2}=1\)? Wait, no, wait. Wait, let's check another pair. Take \((3,8)\) and \((10,15)\). \(x_1 = 3,y_1 = 8,x_2 = 10,y_2=15\). \(m=\frac{15 - 8}{10 - 3}=\frac{7}{7}=1\). Wait, no, wait, \((10,15)\) and \((12,17)\): \(m=\frac{17 - 15}{12 - 10}=\frac{2}{2}=1\). Wait, no, wait, first pair \((1,6)\) and \((3,8)\): \(m=\frac{8 - 6}{3 - 1}=\frac{2}{2}=1\). Wait, but let's check the difference in age and height. From week 1 to week 3 (2 weeks difference), height goes from 6 to 8 (2 cm difference). So rate is \(\frac{2}{2}=1\) cm per week? Wait, no, wait, 2 cm over 2 weeks is 1 cm per week. Wait, let's re - calculate.

Wait, the formula for the rate of change (slope) when dealing with linear relationships (height as a function of age) is \( \text{rate}=\frac{\text{change in height}}{\text{change in age}} \).

For the points \((1,6)\) and \((3,8)\):
Change in height \(=8 - 6 = 2\) cm.
Change in age \(=3 - 1 = 2\) weeks.
Rate \(=\frac{2}{2}=1\) cm per week.

For the points \((3,8)\) and \((10,15)\):
Change in height \(=15 - 8 = 7\) cm.
Change in age \(=10 - 3 = 7\) weeks.
Rate \(=\frac{7}{7}=1\) cm per week.

For the points \((10,15)\) and \((12,17)\):
Change in height \(=17 - 15 = 2\) cm.
Change in age \(=12 - 10 = 2\) weeks.
Rate \(=\frac{2}{2}=1\) cm per week.

So the rate of increase of height with respect to age is 1 cm per week. So option B is correct, A and C are incorrect.

Step2: Determine if age is a function of height or height is a function of age

A relation is a function if for each input, there is exactly one output.

- For "Height is a function of age": For each age (input), there is exactly one height (output). Since it's a linear relationship (given), each age (x - value) has one height (y - value), so height is a function of age (E is correct).

- For "Age is a function of height": Let's see, if height is 6, age is 1. If height is 8, age is 3. If height is 15, age is 10. If height is 17, age is 12. But suppose we have a height value, say, can we have two different ages for the same height? In a linear relationship \(y=mx + b\) (height \(y\), age \(x\)), if we solve for \(x\) in terms of \(y\), \(x=\frac{y - b}{m}\). Since it's a linear function with non - zero slope, it is also a function (because for each \(y\), there is exactly one \(x\)). Wait, but let's check the data. For the given data, each height corresponds to one age. But is this always true? In a linear relationship \(y=x + 5\) (let's find the equation: using point \((1,6)\), \(y=mx + b\), \(6=m(1)+b\), and we found \(m = 1\), so \(b=5\), so \(y=x + 5\) (height \(y\), age \(x\)). Solving for \(x\), \(x=y - 5\). So for each \(y\), there is one \(x\). So age is a function of height (D is correct)? Wait, but wait, let's think again. In a function, for each input, there is exactly one output. If we consider age as a function of height, the input is height (y) and output is age (x). Since the relationship is linear with non - zero slope, it is a function. But wait, let's check the rate again.

Wait, earlier calculation: between \((1,6)\) and \((3,8)\): change in height 2, change in age 2, rate 1. Between \((3,8)\) and \((10,15)\): change in height 7, change in age 7, rate 1. Between \((10,15)\) and \((12,17)\): change in height 2, change in age 2, rate 1. So the rate of increase of height with respect to age is 1 cm per week. So option B is correct (rate is 1 cm per week), E is correct (height is a function of age), and D is correct (age is a function of height)? Wait, no, wait, let's re - examine the function definition.

For height as a function of age: For each age (x), there is exactly one height (y). So it's a function (E is correct). For age as a function of height: For each height (y), there is exactly one age (x). Since the line \(y=x + 5\) is one - to - one (because it has a slope of 1, which is non - zero), it passes the horizontal line test. So age is a function of height (D is correct). Wait, but let's check the rate again.

Wait, the first pair: age increases by 2 (from 1 to 3), height increases by 2 (from 6 to 8). So rate is \(\frac{2}{2}=1\) cm per week. So the rate of height increase with respect to age is 1 cm per week. So option B is correct.

Now, let's analyze each option:

- Option A: Says rate is 2 cm per week. But we calculated rate as 1 cm per week. So A is incorrect.

- Option B: Rate is 1 cm per week. Which matches our calculation. So B is correct.

- Option C: Rate is \(\frac{1}{2}\) cm per week. Incorrect, as our calculation shows 1 cm per week.

- Option D: Age is a function of height. Since for each height value, there is exactly one age value (because the relationship is linear with non - zero slope, so it's one - to - one), so age is a function of height. So D is correct.

- Option E: Height is a function of age. For each age value, there is exactly one height value (since it's a linear relationship), so height is a function of age. So E is correct.

Wait, but let's re - check the slope calculation. Let's use the two points \((1,6)\) and \((12,17)\). Change in height \(=17 - 6 = 11\) cm. Change in age \(=12 - 1 = 11\) weeks. Slope \(=\frac{11}{11}=1\) cm per week. So the rate is 1 cm per week. So B is correct.

And for the function part:

- Height as a function of age: For each age (input), unique height (output) → function (E is correct).

- Age as a function of height: For each height (input), unique age (output) → function (D is correct). Because the linear relationship \(y=x + 5\) (height \(y\), age \(x\)) can be rewritten as \(x=y - 5\), which is a function (each \(y\) gives one \(x\)).

Answer:

B. The height of the plant increases at a constant rate of 1 cm every week.
D. Age is a function of height.
E. Height is a function of age.