Consider the following function. [tex]u(x)=\frac{3}{4 x}[/tex]
Step 2 of 2: Find two points on the graph of this function, other than the origin, that fit within the given [-10,10] by [-10,10] grid. Express each coordinate as an integer or simplified fraction, or round to four decimal places as necessary.
Answer (1)
The problem requires finding two points on the graph of the function u ( x ) = 4 x 3 within the grid [ − 10 , 10 ] × [ − 10 , 10 ] .
Choose x = 1 and calculate u ( 1 ) = 4 ( 1 ) 3 = 0.75 . The point ( 1 , 0.75 ) is valid.
Choose x = 2 and calculate u ( 2 ) = 4 ( 2 ) 3 = 0.375 . The point ( 2 , 0.375 ) is valid.
Two points on the graph are ( 1 , 0.75 ) and ( 2 , 0.375 ) .
Explanation
Understanding the Problem The function given is u ( x ) = 4 x 3 . We need to find two points ( x , u ( x )) on the graph of this function such that − 10 ≤ x ≤ 10 and − 10 ≤ u ( x ) ≤ 10 .
Finding the First Point Let's choose some integer values for x within the given range and calculate the corresponding u ( x ) values. If x = 1 , then u ( 1 ) = 4 ( 1 ) 3 = 4 3 = 0.75 . Since − 10 ≤ 1 ≤ 10 and − 10 ≤ 0.75 ≤ 10 , the point ( 1 , 0.75 ) is a valid point.
Finding the Second Point Now, let's choose another integer value for x . If x = 2 , then u ( 2 ) = 4 ( 2 ) 3 = 8 3 = 0.375 . Since − 10 ≤ 2 ≤ 10 and − 10 ≤ 0.375 ≤ 10 , the point ( 2 , 0.375 ) is also a valid point.
Final Answer Therefore, two points on the graph of the function u ( x ) = 4 x 3 that fit within the given [ − 10 , 10 ] × [ − 10 , 10 ] grid are ( 1 , 0.75 ) and ( 2 , 0.375 ) .
Examples
Understanding functions and their graphs is crucial in many real-world applications. For instance, this function could represent the relationship between the number of workers and the time it takes to complete a task, or the relationship between the price of a product and the demand for it. By finding points on the graph, we can analyze and predict the behavior of these relationships within certain constraints.
