A direct variation function contains the points $(2,14)$ and $(4,28)$. Which equation represents the function?
A. $y=\frac{x}{14}$
B. $y=\frac{x}{7}$
C. $y=7 x$
D. $y=14 x$
Answer (2)
Recognize the direct variation form: y = k x .
Use the point ( 2 , 14 ) to find k : 14 = k × 2 , which gives k = 7 .
Verify using the point ( 4 , 28 ) : 28 = k × 4 , which also gives k = 7 .
Write the equation: y = 7 x .
Explanation
Understanding Direct Variation We are given that a direct variation function contains the points ( 2 , 14 ) and ( 4 , 28 ) . We need to find the equation that represents this function. A direct variation function has the form y = k x , where k is the constant of variation.
Finding the Constant of Variation Let's use the point ( 2 , 14 ) to find the constant of variation k . Substitute x = 2 and y = 14 into the equation y = k x :
14 = k × 2
Solving for k Now, solve for k :
k = 2 14 = 7
Verifying k Let's verify the constant of variation k using the point ( 4 , 28 ) . Substitute x = 4 and y = 28 into the equation y = k x :
28 = k × 4
Confirming k Solve for k :
k = 4 28 = 7 The constant of variation is indeed k = 7 .
Writing the Equation Now, write the equation of the direct variation function by substituting the value of k into the equation y = k x :
y = 7 x
Examples
Direct variation is a relationship between two variables in which one is a constant multiple of the other. For example, the distance you travel at a constant speed varies directly with the time you spend traveling. If you travel at a speed of 60 miles per hour, the equation representing this relationship is d = 60 t , where d is the distance and t is the time. This means that for every hour you travel, the distance increases by 60 miles. Direct variation is also applicable in calculating currency exchange rates, where the amount of money you receive in one currency varies directly with the amount you exchange from another currency.
The function representing the direct variation with points (2,14) and (4,28) is given by the equation y = 7x. This is found by determining the constant of variation k using the points provided. The correct answer is option C, y = 7x.
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