Describe the graph of [tex]${ }^{y=4\lfloor x+2\rfloor}$[/tex] on [tex]$[0,3)$[/tex]. Be specific in your description.
Answer (2)
The function y = 4 ⌊ x + 2 ⌋ is analyzed on the interval [ 0 , 3 ) .
The range of x + 2 is determined to be 2 ≤ x + 2 < 5 .
The possible integer values of ⌊ x + 2 ⌋ are 2, 3, and 4, resulting in y values of 8, 12, and 16, respectively.
The graph consists of horizontal line segments: y = 8 for 0 ≤ x < 1 , y = 12 for 1 ≤ x < 2 , and y = 16 for 2 ≤ x < 3 .
The graph is a step function with steps at y = 8 , y = 12 , and y = 16 .
\boxed{\text{The graph is a step function with steps at } y=8, y=12, \text{ and } y=16.}}
Explanation
Understanding the Problem We want to describe the graph of the function y = 4 ⌊ x + 2 ⌋ on the interval [ 0 , 3 ) . The floor function, denoted by ⌊ x ⌋ , gives the greatest integer less than or equal to x . The graph will consist of horizontal line segments because the floor function produces discrete integer values.
Finding the Range of x+2 First, we need to determine the range of x + 2 when x is in the interval [ 0 , 3 ) . Since 0 ≤ x < 3 , we can add 2 to all parts of the inequality to get 2 ≤ x + 2 < 5 .
Determining Integer Values of the Floor Function Next, we find the possible integer values of ⌊ x + 2 ⌋ based on the range 2 ≤ x + 2 < 5 . The possible integer values are 2, 3, and 4.
Calculating the y Values Now, we determine the corresponding y values for each integer value of ⌊ x + 2 ⌋ :
When ⌊ x + 2 ⌋ = 2 , y = 4 ( 2 ) = 8 . This occurs when 2 ≤ x + 2 < 3 , which means 0 ≤ x < 1 .
When ⌊ x + 2 ⌋ = 3 , y = 4 ( 3 ) = 12 . This occurs when 3 ≤ x + 2 < 4 , which means 1 ≤ x < 2 .
When ⌊ x + 2 ⌋ = 4 , y = 4 ( 4 ) = 16 . This occurs when 4 ≤ x + 2 < 5 , which means 2 ≤ x < 3 .
Describing the Graph Therefore, the graph of y = 4 ⌊ x + 2 ⌋ on the interval [ 0 , 3 ) consists of three horizontal line segments:
y = 8 for 0 ≤ x < 1
y = 12 for 1 ≤ x < 2
y = 16 for 2 ≤ x < 3
Final Description In summary, the graph is a step function with steps at y = 8 , y = 12 , and y = 16 . Each step has a width of 1, and the steps are defined on the intervals [ 0 , 1 ) , [ 1 , 2 ) , and [ 2 , 3 ) respectively.
Examples
Imagine you're designing a simple staircase where each step must be exactly 4 inches high. The function y = 4 ⌊ x + 2 ⌋ models the height y of each step based on its position x . For the first step ( 0 ≤ x < 1 ), the height is 8 inches; for the second step ( 1 ≤ x < 2 ), it's 12 inches; and for the third step ( 2 ≤ x < 3 ), it's 16 inches. This ensures each step maintains the required 4-inch increment, creating a uniform and predictable staircase.
The graph of the function y = 4 ⌊ x + 2 ⌋ on the interval [ 0 , 3 ) consists of horizontal line segments at y = 8 , y = 12 , and y = 16 for the respective intervals [ 0 , 1 ) , [ 1 , 2 ) , and [ 2 , 3 ) . This creates a step function with each step corresponding to the floor function's value. Therefore, the function behaves like a staircase, increasing in defined segments.
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