$y \geq x^2+2$
use $0=x$
$0=y$
What is the $x$-intercept?
What is the $y$-intercept?
Answer (1)
To find the x-intercept, set y = 0 in the inequality = x^2+2"> y " >= x 2 + 2 , resulting in = x^2+2"> 0" >= x 2 + 2 , which has no real solutions.
To find the y-intercept, set x = 0 in the inequality = x^2+2"> y " >= x 2 + 2 , resulting in = 2"> y " >= 2 .
The x-intercept is None.
The y-intercept is 2 .
Explanation
Understanding the Problem We are given the inequality = x^2 + 2"> y " >= x 2 + 2 , and we want to find the x-intercept and y-intercept, given that x = 0 and y = 0 .
Finding the X-intercept To find the x-intercept, we set y = 0 in the inequality. This gives us = x^2 + 2"> 0" >= x 2 + 2 . Rearranging, we get $x^2
Analyzing the X-intercept Since x 2 is always non-negative for any real number x , x 2 cannot be less than or equal to -2. Therefore, there is no real solution for x , and there is no x-intercept.
Finding the Y-intercept To find the y-intercept, we set x = 0 in the inequality. This gives us = 0^2 + 2"> y " >= 0 2 + 2 , which simplifies to = 2"> y " >= 2 .
Analyzing the Y-intercept The y-intercept is the point where the inequality intersects the y-axis. Since = 2"> y " >= 2 , the smallest possible value for y is 2. Therefore, the y-intercept is y = 2 .
Final Answer The x-intercept is None, and the y-intercept is 2.
Examples
Understanding intercepts is crucial in various real-world applications. For example, in economics, the x-intercept of a cost function can represent the break-even point, while the y-intercept represents the fixed costs. Similarly, in physics, intercepts can help determine initial conditions or boundaries of a system. By finding intercepts, we can gain valuable insights into the behavior and characteristics of the system being modeled.
