7. Find the x and y intercepts if [tex]y =\frac{6 x+2}{2 x-4}[/tex]
A. [tex](-1,0)[/tex] and [tex](0,4)[/tex]
B. [tex](-\frac{1}{3}, 0)[/tex] and [tex](0,-\frac{1}{2})[/tex]
C. [tex](-\frac{1}{3}, 3)[/tex] and [tex](0,7)[/tex]
D. [tex](\frac{1}{3}, 0)[/tex] and [tex](0, \frac{1}{2})[/tex]
Answer (1)
To find the x-intercept, set y = 0 and solve for x : 0 = 2 x − 4 6 x + 2 , which gives x = − 3 1 .
To find the y-intercept, set x = 0 and solve for y : y = 2 ( 0 ) − 4 6 ( 0 ) + 2 , which gives y = − 2 1 .
The x-intercept is ( − 3 1 , 0 ) .
The y-intercept is ( 0 , − 2 1 ) .
( − 3 1 , 0 ) and ( 0 , − 2 1 )
Explanation
Problem Analysis We are given the equation y = 2 x − 4 6 x + 2 and asked to find the x and y intercepts.
Finding x-intercept To find the x-intercept, we set y = 0 and solve for x . So we have 0 = 2 x − 4 6 x + 2 . This implies 6 x + 2 = 0 .
Solving for x Solving 6 x + 2 = 0 for x , we get 6 x = − 2 , so x = 6 − 2 = − 3 1 . Thus, the x-intercept is ( − 3 1 , 0 ) .
Finding y-intercept To find the y-intercept, we set x = 0 and solve for y . So we have y = 2 ( 0 ) − 4 6 ( 0 ) + 2 = − 4 2 = − 2 1 . Thus, the y-intercept is ( 0 , − 2 1 ) .
Final Answer Therefore, the x-intercept is ( − 3 1 , 0 ) and the y-intercept is ( 0 , − 2 1 ) . This corresponds to option (b).
Examples
Understanding intercepts is crucial in various real-world applications. For instance, in economics, the x-intercept of a cost function can represent the break-even point, where costs equal revenue. Similarly, in physics, intercepts can indicate initial conditions or points of equilibrium. Knowing how to find intercepts allows us to analyze and interpret data effectively in fields ranging from finance to engineering.
