The function below models the correlation between the number of hours a plant is kept in sunlight $(x)$ and the height $(y)$, in mm, to which it grows: $y=2+4x$
What does the $y$-intercept of this function represent?
A. The original height of the plant was 4 mm.
B. The original height of the plant was 2 mm.
C. The height of the plant increases by 2 mm for every hour of sunlight it receives.
D. The height of the plant increases by 4 mm for every hour of sunlight it receives.
Answer (2)
The problem provides a linear function y = 2 + 4 x that models the height of a plant based on sunlight exposure.
The y -intercept is found by setting x = 0 in the equation.
Substituting x = 0 gives y = 2 + 4 ( 0 ) = 2 .
The y -intercept represents the original height of the plant, which is 2 mm.
Explanation
Understanding the Problem The given function is y = 2 + 4 x , which models the height ( y ) of a plant in mm based on the number of hours ( x ) it is kept in sunlight. We need to determine what the y -intercept of this function represents in the context of the problem.
Finding the y-intercept The y -intercept is the value of y when x = 0 . In this context, it represents the height of the plant when it receives 0 hours of sunlight, which is the original height of the plant.
Calculating the y-intercept To find the y -intercept, we substitute x = 0 into the equation: y = 2 + 4 ( 0 ) = 2 + 0 = 2 So, the y -intercept is 2.
Interpreting the y-intercept The y -intercept of 2 means that when the plant receives 0 hours of sunlight, its height is 2 mm. Therefore, the original height of the plant was 2 mm.
Examples
Understanding the y-intercept is crucial in various real-life scenarios. For instance, in a business model, if y represents the total cost and x represents the number of units produced, the y-intercept would represent the fixed costs, which are the costs incurred even when no units are produced. Similarly, in a physics experiment, if y represents the distance traveled and x represents time, the y-intercept could represent the initial position of an object. Recognizing and interpreting the y-intercept helps in making informed decisions and understanding initial conditions.
The y -intercept of the function y = 2 + 4 x is 2 mm, which represents the original height of the plant when it receives 0 hours of sunlight. Thus, the correct answer is B. The original height of the plant was 2 mm.
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