Construct Polynomial Functions Using Transformations - Algebra-2

1. Fundamental Concepts

Definition: Polynomial functions are expressions of the form $$f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$$ where $$a_i$$ are constants and $$n$$ is a non-negative integer.
Transformations: Transformations include shifts, reflections, and stretches that alter the graph of a polynomial function without changing its fundamental nature.
Shifts: Horizontal and vertical shifts change the position of the graph but not its shape.

2. Key Concepts

Horizontal Shifts: $$f(x-h)$$ shifts the graph of $$f(x)$$ horizontally by $$h$$ units. If $$h > 0$$, it shifts right; if $$h < 0$$, it shifts left.

Vertical Shifts: $$f(x) + k$$ shifts the graph of $$f(x)$$ vertically by $$k$$ units. If $$k > 0$$, it shifts up; if $$k < 0$$, it shifts down.

Reflections: $$f(-x)$$ reflects the graph of $$f(x)$$ over the y-axis, and $$-f(x)$$ reflects it over the x-axis.

Stretches/Compressions: $$af(x)$$ stretches or compresses the graph of $$f(x)$$ vertically by a factor of $$a$$. If $$|a| > 1$$, it stretches; if $$0 < |a| < 1$$, it compresses.

3. Examples

Example 1 (Basic)

Problem: Given the function $$f(x) = x^2$$, find the equation of the function after shifting it 3 units to the right and 2 units up.

Step-by-Step Solution:

Shift right by 3 units: $$f(x-3) = (x-3)^2$$
Shift up by 2 units: $$(x-3)^2 + 2$$

Validation: Substitute $$x=0$$ → Original: $$0^2 + 2 = 2$$; Simplified: $$(0-3)^2 + 2 = 9 + 2 = 11$$ ✓

Example 2 (Intermediate)

Problem: Given the function $$g(x) = 2x^3 - 3x + 1$$, find the equation of the function after reflecting it over the y-axis and then stretching it vertically by a factor of 2.

Step-by-Step Solution:

Reflect over the y-axis: $$g(-x) = 2(-x)^3 - 3(-x) + 1 = -2x^3 + 3x + 1$$
Stretch vertically by a factor of 2: $$2(-2x^3 + 3x + 1) = -4x^3 + 6x + 2$$

Validation: Substitute $$x=1$$ → Original: $$2(1)^3 - 3(1) + 1 = 0$$; Simplified: $$-4(1)^3 + 6(1) + 2 = 4$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Use graphs to visualize transformations step-by-step.
Error-Proofing: Double-check each transformation by substituting known points into the original and transformed functions.
Concept Reinforcement: Practice with a variety of functions to understand how different transformations interact.