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函数的变换
函数 $y = g(x) = af(x-b) + c$ 的图像与 $y = f(x)$ 的图像有何关系？
1. Introduction
Introduction
2. Translations of Functions
Translations of Functions
3. Vertical stretches and reflections
Vertical stretches and reflections
4. Combining shifts and stretches: why order sometimes matters
Combining shifts and stretches: why order sometimes matters
5. Summary
Summary
6. Vertical shift of a parent parabola
Starting from f(x) = x^2, what does g(x) = f(x) + a do to the graph as a varies?
7. Combining a vertical and a horizontal shift
Relative to r, how do g(x) = r(x) + 2, h(x) = r(x + 1) and f(x) = r(x + 1) + 2 move the point (2, -1)?
8. Translating a quadratic
Starting from q(x) = x^2, what is p(x) = q(x + 3) - 4, and how is it a transformation?
9. Vertical stretch and compression
For v(x) = c f(x), how does c affect the graph, and what happens to (-2, 1) under 3r and (1/3)r?
10. Reflection and compression together
How is m(x) = 2r(x + 1) - 1 obtained from r, and where does the point (2, -1) land?
11. A transformed point
For p(x) = -(1/2)f(x - 1) + 2, the point (-2, 2) on f maps to a new point. What is its y-coordinate?
12. Comparing two transformation formulas
Are p(x) = -(1/2)f(x-1) + 2 and r(x) = (1/2)(-f(x-1) - 4) the same function?
13. Building a transformation formula
For g, apply a shift 1.25 left, reflection across the x-axis with a vertical stretch of 2.5, then a shift up 1.75. What is the coefficient multiplying g(x + 1.25)?
14. Vertical shifts leave average rates unchanged
For f(x) = x^2 and g(x) = f(x) + 5, how do their average rates on [-3, -1] compare?
15. Reading a point-slope line as a transformation
How is L(x) = -4(x - 3) + 5 built from the parent f(x) = x?
16. Multiplying the input
Starting from f(x) = (x - 2)^2 + 1, how does g(x) = f(4x) differ from f?
17. An inside change shifts the graph horizontally
Replacing x with x + 3 inside a function shifts its graph three units to the right.