Graph of y = -x² + 4
Interactive graph of y = -x² + 4 (inverted parabola). Explore the downward-opening parabola, its maximum value, and x-intercepts.
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Understanding the Function
The function y = -x² + 4 is a downward-opening parabola with its vertex at (0, 4). The negative coefficient on x² flips the standard parabola upside down, making the vertex a maximum point rather than a minimum. The curve crosses the x-axis at x = -2 and x = 2.
Inverted parabolas model many real-world situations where there is an optimal peak value: the trajectory of a projectile (maximum height), revenue as a function of price (profit maximization), or the intensity distribution of a laser beam. Setting y = 0 and solving gives the roots x = +/- 2.
Key properties: vertex at (0, 4), axis of symmetry at x = 0, x-intercepts at (-2, 0) and (2, 0), y-intercept at (0, 4), opens downward, maximum value of 4. The domain is all real numbers, and the range is (-infinity, 4].