Raphael graphed the functions [tex]g(x)=x+2[/tex] and [tex]f(x)=x-1[/tex]. How many units below the [tex]y[/tex]-intercept of [tex]g(x)[/tex] is the [tex]y[/tex] intercept of [tex]f(x)[/tex]?
Answer (2)
Find the y-intercept of g ( x ) by setting x = 0 : g ( 0 ) = 2 .
Find the y-intercept of f ( x ) by setting x = 0 : f ( 0 ) = − 1 .
Calculate the difference between the y-intercepts: 2 − ( − 1 ) = 3 .
The y-intercept of f ( x ) is 3 units below the y-intercept of g ( x ) , so the answer is 3 .
Explanation
Understanding the Problem We are given two linear functions, g ( x ) = x + 2 and f ( x ) = x − 1 . We need to find the difference between their y -intercepts. The y -intercept is the point where the graph of the function intersects the y -axis, which occurs when x = 0 .
Finding the y-intercept of g(x) To find the y -intercept of g ( x ) , we set x = 0 and evaluate g ( 0 ) : g ( 0 ) = 0 + 2 = 2 So, the y -intercept of g ( x ) is 2.
Finding the y-intercept of f(x) To find the y -intercept of f ( x ) , we set x = 0 and evaluate f ( 0 ) :
f ( 0 ) = 0 − 1 = − 1 So, the y -intercept of f ( x ) is -1.
Calculating the Difference Now, we need to find how many units below the y -intercept of g ( x ) is the y -intercept of f ( x ) . This is the difference between the y -intercepts: g ( 0 ) − f ( 0 ) = 2 − ( − 1 ) = 2 + 1 = 3 Therefore, the y -intercept of f ( x ) is 3 units below the y -intercept of g ( x ) .
Examples
Understanding y-intercepts is crucial in many real-world applications. For example, in a linear cost function C ( x ) = m x + b , where x is the number of units produced, b represents the fixed costs (the y-intercept). If two companies have cost functions C 1 ( x ) = 2 x + 5 and C 2 ( x ) = 2 x + 2 , the difference in their fixed costs is 5 − 2 = 3 . This means the first company has $3 more in fixed costs than the second company, regardless of the number of units they produce. Analyzing y-intercepts helps in comparing initial values or fixed components in linear models.
The y-intercept of g ( x ) = x + 2 is 2, while the y-intercept of f ( x ) = x − 1 is -1. Therefore, the y-intercept of f ( x ) is 3 units below the y-intercept of g ( x ) . The answer is 3 units.
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