Graph of y = x² - 4
Interactive graph of y = x² - 4 (vertically shifted parabola). Explore its roots, vertex below the x-axis, and factoring as a difference of squares.
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Understanding the Function
The function y = x² - 4 is a parabola shifted 4 units downward from the standard y = x². Its vertex is at (0, -4), and it crosses the x-axis at x = -2 and x = 2. The expression factors as (x - 2)(x + 2), a classic difference of squares.
This function is a key example in algebra for understanding vertical translations of graphs and for practicing factoring techniques. The difference of squares pattern a² - b² = (a - b)(a + b) is one of the most frequently used factoring identities, appearing in everything from simplifying fractions to rationalizing denominators.
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 upward, minimum value of -4. The domain is all real numbers, and the range is [-4, infinity). The discriminant is 16, confirming two distinct real roots.