Answer :
Let's analyze the given function step-by-step:
### Given Function:
The function provided to model the relationship between the number of hours a plant is kept in sunlight [tex]\((x)\)[/tex] and the height [tex]\((y)\)[/tex] in millimeters is:
[tex]\[ y = 4 + 2x \][/tex]
### Form of the Function:
This function is in the linear form:
[tex]\[ y = mx + c \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( c \)[/tex] is the y-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0).
### Components from the Given Function:
1. y-Intercept ([tex]\( c \)[/tex]): From the given function [tex]\( y = 4 + 2x \)[/tex], we identify that [tex]\( c = 4 \)[/tex]. This means that when the plant is not exposed to any sunlight ([tex]\( x = 0 \)[/tex]), the height of the plant is [tex]\( 4 \)[/tex] mm. Thus, the original height of the plant is [tex]\( 4 \)[/tex] mm.
2. Slope ([tex]\( m \)[/tex]): From the function, we see that the coefficient of [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex]. This coefficient represents the slope [tex]\( m \)[/tex]. The slope indicates how much [tex]\( y \)[/tex] changes for a unit change in [tex]\( x \)[/tex]. Specifically, it tells us the rate of change of the plant's height with respect to the amount of sunlight received.
### Interpretation of the Slope:
- The slope ([tex]\( m = 2 \)[/tex]) means that for every additional hour of sunlight ([tex]\( x \)[/tex]), the plant's height ([tex]\( y \)[/tex]) increases by [tex]\( 2 \)[/tex] mm.
- In simpler terms, the plant's height increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives.
### Correct Interpretation:
Based on the given options and our analysis:
1. "The original height of the plant was [tex]\( 4 \)[/tex] mm." This is correct, but it does not address the slope.
2. "The original height of the plant was [tex]\( 2 \)[/tex] mm." This is incorrect.
3. "The height of the plant increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives." This accurately reflects the meaning of the slope.
4. "The height of the plant increases by [tex]\( 4 \)[/tex] mm for every hour of sunlight it receives." This is incorrect.
### Conclusion:
The slope of the function [tex]\( y = 4 + 2x \)[/tex] represents that "The height of the plant increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives."
Thus, the correct option is:
The height of the plant increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives.
### Given Function:
The function provided to model the relationship between the number of hours a plant is kept in sunlight [tex]\((x)\)[/tex] and the height [tex]\((y)\)[/tex] in millimeters is:
[tex]\[ y = 4 + 2x \][/tex]
### Form of the Function:
This function is in the linear form:
[tex]\[ y = mx + c \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( c \)[/tex] is the y-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0).
### Components from the Given Function:
1. y-Intercept ([tex]\( c \)[/tex]): From the given function [tex]\( y = 4 + 2x \)[/tex], we identify that [tex]\( c = 4 \)[/tex]. This means that when the plant is not exposed to any sunlight ([tex]\( x = 0 \)[/tex]), the height of the plant is [tex]\( 4 \)[/tex] mm. Thus, the original height of the plant is [tex]\( 4 \)[/tex] mm.
2. Slope ([tex]\( m \)[/tex]): From the function, we see that the coefficient of [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex]. This coefficient represents the slope [tex]\( m \)[/tex]. The slope indicates how much [tex]\( y \)[/tex] changes for a unit change in [tex]\( x \)[/tex]. Specifically, it tells us the rate of change of the plant's height with respect to the amount of sunlight received.
### Interpretation of the Slope:
- The slope ([tex]\( m = 2 \)[/tex]) means that for every additional hour of sunlight ([tex]\( x \)[/tex]), the plant's height ([tex]\( y \)[/tex]) increases by [tex]\( 2 \)[/tex] mm.
- In simpler terms, the plant's height increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives.
### Correct Interpretation:
Based on the given options and our analysis:
1. "The original height of the plant was [tex]\( 4 \)[/tex] mm." This is correct, but it does not address the slope.
2. "The original height of the plant was [tex]\( 2 \)[/tex] mm." This is incorrect.
3. "The height of the plant increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives." This accurately reflects the meaning of the slope.
4. "The height of the plant increases by [tex]\( 4 \)[/tex] mm for every hour of sunlight it receives." This is incorrect.
### Conclusion:
The slope of the function [tex]\( y = 4 + 2x \)[/tex] represents that "The height of the plant increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives."
Thus, the correct option is:
The height of the plant increases by [tex]\( 2 \)[/tex] mm for every hour of sunlight it receives.
