Determine the [tex]$x$[/tex]- and [tex]$y$[/tex]-intercepts for the graph defined by the given equation: [tex]$y=x+8$[/tex]
A. [tex]$x$[/tex]-intercept is [tex]$(0,8)$[/tex], [tex]$y$[/tex]-intercept is [tex]$(-8,0)$[/tex]
B. [tex]$x$[/tex]-intercept is [tex]$(0,-8)$[/tex], [tex]$y$[/tex]-intercept is [tex]$(8,0)$[/tex]
C. [tex]$x$[/tex]-intercept is [tex]$(-8,0)$[/tex], [tex]$y$[/tex]-intercept is [tex]$(0,8)$[/tex]
D. [tex]$x$[/tex]-intercept is [tex]$(8,0)$[/tex], [tex]$y$[/tex]-intercept is [tex]$(0,-8)$[/tex]
Please select the best answer from the choices provided.
Answer (2)
The x-intercept for the given equation is (-8, 0) and the y-intercept is (0, 8). Therefore, the correct choice among the provided options is C. This shows how to find the x- and y-intercepts of a linear equation.
;
To find the x -intercept, set y = 0 and solve for x : 0 = x + 8 , so x = − 8 .
To find the y -intercept, set x = 0 and solve for y : y = 0 + 8 , so y = 8 .
The x -intercept is ( − 8 , 0 ) and the y -intercept is ( 0 , 8 ) .
The correct answer is C: x -intercept is ( − 8 , 0 ) , y -intercept is ( 0 , 8 ) .
Explanation
Understanding the Problem We are given the equation y = x + 8 and need to find the x - and y -intercepts. The x -intercept is the point where the graph crosses the x -axis (where y = 0 ), and the y -intercept is the point where the graph crosses the y -axis (where x = 0 ).
Finding the x-intercept To find the x -intercept, we set y = 0 in the equation y = x + 8 :
0 = x + 8 Solving for x , we subtract 8 from both sides: x = − 8 So the x -intercept is the point ( − 8 , 0 ) .
Finding the y-intercept To find the y -intercept, we set x = 0 in the equation y = x + 8 :
y = 0 + 8 y = 8 So the y -intercept is the point ( 0 , 8 ) .
Final Answer Therefore, the x -intercept is ( − 8 , 0 ) and the y -intercept is ( 0 , 8 ) . Comparing this to the given options, we see that option C matches our result.
Examples
Understanding intercepts is crucial in many real-world applications. For example, in business, the x-intercept of a cost function can represent the break-even point, where costs equal revenue. Similarly, in physics, the y-intercept of a velocity-time graph can represent the initial velocity of an object. Knowing how to find and interpret intercepts allows us to analyze and make predictions about various phenomena.
