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How does the value of a in the function affect its graph when compared to the graph of the quadratic parent function?
g left parenthesis x right parenthesisg(x)equals=minus−66xsquared2
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It looks like the function is g(x) = -6x^2 (please tell me if that was a typo). In general, for g(x) = a x^2 compared to the parent f(x) = x^2:
- Sign of a: if a > 0 the parabola opens upward (same as f); if a < 0 it opens downward (a reflection across the x-axis).
- Magnitude of a: if |a| > 1 the graph is a vertical stretch (narrower) by factor |a|; if 0 < |a| < 1 it is a vertical compression (wider) by factor 1/|a|.
- The vertex and axis of symmetry stay the same as the parent (vertex at (0,0), axis x = 0) for no horizontal/vertical shifts.
Applied to g(x) = -6x^2 (a = -6): the parabola opens downward (negative sign) and is vertically stretched by a factor of 6, so it is much narrower than y = x^2. For example, the point (1,1) on y = x^2 becomes (1, −6) on g(x).
- Sign of a: if a > 0 the parabola opens upward (same as f); if a < 0 it opens downward (a reflection across the x-axis).
- Magnitude of a: if |a| > 1 the graph is a vertical stretch (narrower) by factor |a|; if 0 < |a| < 1 it is a vertical compression (wider) by factor 1/|a|.
- The vertex and axis of symmetry stay the same as the parent (vertex at (0,0), axis x = 0) for no horizontal/vertical shifts.
Applied to g(x) = -6x^2 (a = -6): the parabola opens downward (negative sign) and is vertically stretched by a factor of 6, so it is much narrower than y = x^2. For example, the point (1,1) on y = x^2 becomes (1, −6) on g(x).
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