Timmy writes the equation [tex]f(x)=\frac{1}{4} x-1[/tex]. He then doubles both of the terms on the right side to create the equation [tex]g(x)=\frac{1}{2} x-2[/tex]. How does the graph of [tex]g(x)[/tex] compare to the graph of [tex]f(x)[/tex]?
A. The line of [tex]g(x)[/tex] is steeper and has a higher [tex]y[/tex]-intercept.
B. The line of [tex]g(x)[/tex] is less steep and has a lower [tex]y[/tex]-intercept.
C. The line of [tex]g(x)[/tex] is steeper and has a lower [tex]y[/tex]-intercept.
D. The line of [tex]g(x)[/tex] is less steep and has a higher [tex]y[/tex]-intercept.
Answer (1)
The slope of f ( x ) = 4 1 x − 1 is 4 1 and its y-intercept is − 1 .
The slope of g ( x ) = 2 1 x − 2 is 2 1 and its y-intercept is − 2 .
Since \frac{1}{4}"> 2 1 > 4 1 , g ( x ) is steeper than f ( x ) .
Since − 2 < − 1 , g ( x ) has a lower y-intercept than f ( x ) .
The line of g ( x ) is steeper and has a lower y -intercept. \boxed{The line of g(x) i ss t ee p er an d ha s a l o w er y$-intercept.}
Explanation
Analyzing the Equations We are given two linear equations: f ( x ) = 4 1 x − 1 and g ( x ) = 2 1 x − 2 . We need to compare their graphs by analyzing their slopes and y-intercepts.
Identifying Slope and Y-intercept of f(x) For the equation f ( x ) = 4 1 x − 1 , the slope is 4 1 and the y-intercept is − 1 .
Identifying Slope and Y-intercept of g(x) For the equation g ( x ) = 2 1 x − 2 , the slope is 2 1 and the y-intercept is − 2 .
Comparing Slopes Comparing the slopes, we see that \frac{1}{4}"> 2 1 > 4 1 , so the graph of g ( x ) is steeper than the graph of f ( x ) .
Comparing Y-intercepts Comparing the y-intercepts, we see that − 2 < − 1 , so the graph of g ( x ) has a lower y-intercept than the graph of f ( x ) .
Conclusion Therefore, the line of g ( x ) is steeper and has a lower y -intercept compared to the graph of f ( x ) .
Examples
Understanding the slopes and y-intercepts of linear equations helps us predict how one quantity changes in relation to another. For example, if f ( x ) represents the cost of producing x items at one factory and g ( x ) represents the cost at another factory, we can quickly see which factory has a higher initial cost (y-intercept) and which factory's cost increases more rapidly with each item produced (slope). This helps in making informed business decisions.
