State the intercepts, domain and range for y= |0.5x^2-2|.
The graph should make things clear:
http://www.wolframalpha.com/input/?i=%7Cx%5E2%2F2-2%7C
To find the intercepts, domain, and range of the function y = |0.5x^2 - 2|, we can break it down step by step.
1. Intercepts:
- The x-intercepts are the points where the graph of the equation intersects the x-axis. To find these points, set y = 0 and solve for x.
0 = |0.5x^2 - 2|
Taking the absolute value, we get:
0 = 0.5x^2 - 2
2 = 0.5x^2
x^2 = 4
x = ±2
So, the x-intercepts are x = -2 and x = 2.
- The y-intercept is the point where the graph intersects the y-axis. To find this point, set x = 0 and solve for y.
y = |0.5(0)^2 - 2|
y = |-2|
y = 2
So, the y-intercept is y = 2.
2. Domain:
The domain of a quadratic equation is the set of all possible x-values for which the equation is defined. Since the equation is a quadratic function with no restrictions, the domain is all real numbers, (-∞, +∞).
3. Range:
The range of the function y = |0.5x^2 - 2| is the set of all possible y-values. To determine the range, consider that the absolute value of a number is always positive or zero. Therefore, the range is y ≥ 0, which means all non-negative real numbers, [0, +∞).
To summarize:
- X-intercepts: -2 and 2
- Y-intercept: 2
- Domain: (-∞, +∞)
- Range: [0, +∞)