Asked by Ian
What is the domain, range, X-intercept, Y-intercept, relative extrema, end behavior, and the type of parent function of y=4^3+x^2-2x+2.
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Answered by
GPT-5 mini
AI
Interpreting the expression as y = 4^3 + x^2 − 2x + 2 = x^2 − 2x + 66 (since 4^3 = 64):
- Domain: all real numbers, (−∞, ∞).
- Rewrite (vertex form): y = (x − 1)^2 + 65.
- Vertex (relative and absolute extremum): (1, 65). This is a minimum (global minimum).
- Range: y ≥ 65, i.e. [65, ∞).
- X-intercepts: none (no real roots; discriminant = 4 − 264 = −260 < 0).
- Y-intercept: y(0) = 66 → (0, 66).
- Axis of symmetry: x = 1.
- End behavior: as x → ±∞, y → +∞ (both ends up).
- Parent function: quadratic, y = x^2 (parabola opening upward).
- Domain: all real numbers, (−∞, ∞).
- Rewrite (vertex form): y = (x − 1)^2 + 65.
- Vertex (relative and absolute extremum): (1, 65). This is a minimum (global minimum).
- Range: y ≥ 65, i.e. [65, ∞).
- X-intercepts: none (no real roots; discriminant = 4 − 264 = −260 < 0).
- Y-intercept: y(0) = 66 → (0, 66).
- Axis of symmetry: x = 1.
- End behavior: as x → ±∞, y → +∞ (both ends up).
- Parent function: quadratic, y = x^2 (parabola opening upward).
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