Pre-Calculus: Functions, Symmetry, and Transformations

Function used in example: $f(x) = 2x^3 - 3x + 5.$
Given inputs: $a = -1, \quad b = 2.$
Evaluate the function at the given inputs:
- $f(-1) = 2(-1)^3 - 3(-1) + 5 = -2 + 3 + 5 = 6.$
- $f(2) = 2(2)^3 - 3(2) + 5 = 16 - 6 + 5 = 15.$
Coordinates on the graph:
- $(-1, 6), \quad (2, 15).$
Average rate of change (arc between a and b):
- $\text{ARC} = \frac{f(b) - f(a)}{b - a} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{15 - 6}{3} = 3.$
Interpretation:
- The average rate of change over the interval is the slope of the secant line connecting the two points.
Connection to calculus:
- As the distance between the two x-values shrinks toward zero, the average rate of change approaches the instantaneous slope at a point (the derivative). This is the first idea of calculus: instantaneous slopes.

Even functions:
- Definition: $f(-x) = f(x)\quad\text{for all } x,$ which means symmetry about the y-axis.
Odd functions:
- Definition: $f(-x) = -f(x)\quad\text{for all } x,$ which means symmetry about the origin.
Visual interpretation:
- Even: symmetry with respect to the y-axis.
- Odd: symmetry with respect to the origin; the point ((x, f(x))) maps to ((-x, -f(x))).
Important caution:
- Do not confuse multiplying the entire function by (-1) with the test for oddness. The test is on the relation between (f(-x)) and (f(x)):
- If $f(-x) = f(x)$ , the function is even.
- If $f(-x) = -f(x)$ , the function is odd.
Mnemonic to remember oddness:
- Odd begins with an O, origin symmetry begins with an O, and any negative inside moves to outside (inside to outside), all starting with O.
Example to illustrate oddness:
- If $f(x) = 4x^3 - 2x,$ then
- $f(-x) = 4(-x)^3 - 2(-x) = -4x^3 + 2x = -(4x^3 - 2x) = -f(x),$
- hence the function is odd.
caveat: some functions can be neither even nor odd.
Quick practice idea from the transcript (polyfunction test): for a function like $h(x) = 4x^3 - 2x,$ compute $h(-x)$ and compare to $h(x)\ (\text{even})\,\text{or}\, -h(x)\ (\text{odd}).$

Definitions:
- Increasing on an interval: the graph goes up as the input x increases to the right.
- Decreasing on an interval: the graph goes down as x increases to the right.
- Constant on an interval: the graph is horizontal (slope 0) on that interval.
Key idea:
- A function can have multiple subintervals with different behavior (increasing, decreasing, or constant).
How to identify:
- Look at the sign of the slope on each interval:
- Positive slope => increasing.
- Negative slope => decreasing.
- Zero slope => constant.
Relation to instantaneous slope:
- The same concept as instantaneous slope in calculus, viewed over intervals rather than at a single point.

Relative (local) maximum:
- A peak that is higher than nearby points.
Relative (local) minimum:
- A valley that is lower than nearby points.
Distinction from absolute extrema:
- Relative maxima/minima are not necessarily the global (absolute) max/min over the entire domain.
Visual intuition:
- A wavy function may have several bumps (relative maxima) and valleys (relative minima).

Idea:
- A parent function is a basic form from which many transformed functions are derived.
- Transformations include translation, reflection, and scaling (stretch/shrink).
Two reference points per parent help compare originals and transformed versions.
List of common parent functions and their basic form:
- Constant: $f(x) = c$
- Horizontal line; every input yields the same output; still a function.
- Identity: $f(x) = x$
- Points: ((0,0)), ((1,1)); base form of any linear function.
- Square (Quadratic): $f(x) = x^2$
- Parabola opening upward; points: ((0,0)), ((1,1)).
- Cubic: $f(x) = x^3$
- S-shaped; grows negative on the left, positive on the right; points: ((0,0)), ((1,1)).
- Reciprocal: $f(x) = \frac{1}{x}$
- Two branches; asymptote at (x = 0); typical reference: ((1,1)) is a point; ((0,0)) is not on the graph but can be used as a center for thinking about symmetry.
- Absolute Value: $f(x) = |x|$
- V-shaped; piecewise: $f(x) = \begin{cases}-x,& x<0 \\ x,& x\ge 0\end{cases}$ ; reference: ((0,0)).
- Square Root: $f(x) = \sqrt{x}$
- Domain: (x \ge 0); one-branch curve in the first quadrant.
Note:
- These parent functions form the basis for many transformed functions.

Core idea:
- Transformations modify a base function to produce new graphs by altering input or output values.
- Always use function notation to describe the operation (e.g., f(x)).
Translations (shifts):
- Vertical translations:
- Up by c: $f(x) \mapsto f(x) + c.$
- Down by c: $f(x) \mapsto f(x) - c.$
- Horizontal translations:
- Right by c: replacing the input by (x - c) yields the transformed function $f(x - c).$
- Left by c: replacing the input by (x + c) yields $f(x + c).$
Important intuition:
- The inside of the function controls left/right movement; the outside controls up/down movement. The signs can feel counterintuitive, e.g., inside shifts correspond to opposite directional movement.
How to detect stretch/shrink and reflections (brief preview):
- To detect vertical/horizontal stretching or compression, compare distances between corresponding points before and after the transformation.
- If horizontal distances change while vertical distances scale, you may have horizontal stretch/compression (and possibly a reflection in some cases).
Notes for later material:
- Reflections and more advanced transformations follow, but the basic ideas above are the foundation for analyzing transformed functions.

Function notation recap:
- A function assigns a unique output to each input; the vertical line test confirms it.
Key ideas from the transcript:
- Average rate of change is the slope of the secant line between two points on the graph.
- Instantaneous slope is the limit of the average rate of change as the interval approaches zero.
- Symmetry concepts (even and odd) help classify functions through algebraic tests.
- Parent functions provide building blocks for a wide variety of functions.
- Transformations (translations, reflections, and scaling) derive many functions from a parent.