Notes on Transformation of Functions: Piecewise Functions and Graph Transformations

Piecewise Functions

• Definition: A piecewise function is a function that is separated into multiple pieces, each valid on its own sub-domain of x.
• General form (example from lecture):
  $f(x)=\begin{cases}<br /> x,&amp; x<2\ x^2,& 2\le x\le 4\ 3x+5,& x>4<br /> \end{cases}$
• How to read it:
  • For x<2, use the expression f(x) = x.
  • For 2≤x≤4, use f(x) = x^2.
  • For x>4, use f(x) = 3x+5.
• Evaluating specific values (from the lecture example):
  • f(5): since 5>4, use the third piece: $f(5)=3\cdot5+5=20.$
  • f(2): since 2 falls in the second piece (2≤x≤4), $f(2)=2^2=4.$
  • Note about the open/closed endpoints:
  • The first piece applies only for x<2, so at x=2 the first piece does not apply (open circle at (2,2) on the first piece).
  • The second piece applies at x=2, so the point (2,4) is part of the graph.
  • The third piece applies for x>4, so at x=4 the third piece does not apply; the value there comes from the second piece (f(4)=4^2=16).
  • The endpoint at x=4 for the third piece would have y=3(4)+5=17, but since x>4, that point is not included; you would see an open circle at (4,17) on the third piece if drawn separately.
• Graphing approach (two equivalent methods discussed): 1) Graph each piece separately and then erase the parts that are not in the domain of that piece.
  • For the line piece y = x: only where x<2; this yields an open circle at (2,2).
  • For the parabola y = x^2: between x=2 and x=4, including endpoints, producing the segment from (2,4) to (4,16).
  • For the line piece y = 3x+5: for x>4; begins just to the right of x=4, approaching (4,17) but not including it.
  • This method yields a continuous-looking curve made of three pieces meeting at the breakpoints with appropriate open/closed circles.
    2) Evaluate the boundary values to plot the join points, then sketch:
  • At x=2: from the first piece, y=2 (open circle); from the second piece, y=4 (included). Plot (2,4) as the actual graph point.
  • At x=4: from the second piece, y=16 (included); from the third piece, the value would be 17 at x=4 but x=4 is not allowed for that piece (open circle at (4,17) if drawn separately).
  • The parabolic segment runs from (2,4) to (4,16).
  • The third piece starts just right of x=4 along y=3x+5, beginning near (4+,17).
• Quick conclusion: the two drawing methods produce the same piecewise graph; you can choose the one you find easier.
• Practice prompts given in the lecture:
  • Graph f(x) and g(x) (definitions not explicitly written in this excerpt) and evaluate: f(5), f(-2), g(0), g(3), f(0).
  • Note about attendance and class logistics were mentioned during the activity (census day / roll call). The instructor paused to take roll: Saif, Allison, Emily, Owen, Julian, Eduardo, Alexis, Maya, Gianna, Analise, Kubra, Kalyana, Preston, Ella, Ava, Moises, etc.

Graphing transformations: overview of core rules

• Goal: transform a parent function f(x) by shifts, reflections, and scalings to visualize the graph of transformed function.
• Base idea: apply simple, well-defined operations to the parent graph of f(x).
• Six core rules (as presented):
  • Rule 1: Vertical shifts (outside the function):
  • If you see f(x) + k, shift the entire graph up by k: the y-values increase by k.
  • If you see f(x) - k, shift the entire graph down by k: the y-values decrease by k.
  • Rule 2: Horizontal shifts (inside the function):
  • If you see f(x + k), shift the graph left by k (the input is increased by k, so the graph moves left).
  • Example intuition: with f(x) = x^2, f(x+2) is the same parabola moved left by 2 units (the point that was at (0,0) moves to (-2,0)).
  • Rule 3: Reflection across the y-axis (inside negative):
  • If you see f(-x), the graph is reflected across the y-axis.
  • Example: f(x) = x^2 remains x^2 after f(-x); f(x) = x^3 becomes (-x)^3 = -x^3, which is the reflected shape across the y-axis.
  • Rule 4: Reflection across the x-axis (outside negative):
  • If you see -f(x), the graph is reflected across the x-axis (flip vertically).
  • Example: f(x) = x^2 becomes -x^2 when transformed to -f(x); the parabola opens downward.
  • Rule 5: Vertical stretch/compression (outside multiplier a):
  • If a > 1, the graph is vertically stretched (taller) by factor a.
  • If 0 < a < 1, the graph is vertically compressed (wider) by factor a.
  • If a < 0, the graph is reflected across the x-axis and vertically stretched by |a|.
  • Rule 6: Horizontal stretch/compression (inside multiplier b in f(bx))
  • If b > 1, the graph is horizontally compressed by factor 1/b.
  • If 0 < b < 1, the graph is horizontally stretched by factor 1/b.
  • Note on mental model: separate the two “stretch/compression” ideas for ease of visualization; use x-y tables when dealing with complex scaling.

Worked examples of transformation rules

• Example 1: f(x) = (1/3) x^2 (compression, since the outside factor is between 0 and 1)
  • Build an x-y table (sample points):
  • x = -2: y = (1/3)(-2)^2 = (1/3)(4) = 4/3
  • x = -1: y = (1/3)(-1)^2 = 1/3
  • x = 0: y = 0
  • x = 1: y = (1/3)(1)^2 = 1/3
  • x = 2: y = (1/3)(4) = 4/3
  • Graph becomes wider than the parent x^2 due to compression (height is reduced by factor 3, but the width expands visually).
• Example 2: f(x) = x^3 with inner negation (f(-x)) and outer scaling variants:
  • Inside: replace x by -x -> reflection across the y-axis.
  • Example transformation: replacing inside with -x for f(x) = x^3 yields f(-x) = (-x)^3 = -x^3, which is the reflection across the y-axis of the original cubic shape.
  • If you instead multiply outside (e.g., -f(x) or a different outside a), you would also reflect across the x-axis in addition to any vertical scaling.
• Example 3: Combined inside and outside transformations, such as f(-2x) or f(0.5x):
  • For f(-2x): horizontal compression by factor 1/2 (since b = -2, |b| > 1) and reflection across the y-axis from the negative sign inside.
  • For f(0.5x): horizontal stretch by factor 2 (since 0 < 0.5 < 1).
• Example 4: A concrete composition (from the lecture): determine the graph of g(x) = f(-x - 1) + 2 when f is a known parent function (diagrammatic steps discussed).
  • Interpret as: shift left by 1 (due to -x-1 inside), then reflect across the y-axis if applicable, then shift up by 2 (outside). The exact order of applying shifts vs reflections should be tracked as per standard rules.

Even and odd functions

• Definitions:
  • Even function: f(-x) = f(x) for all x in the domain.
  • Odd function: f(-x) = -f(x) for all x in the domain.
• How to check:
  • Compute f(-x) and compare it to f(x).
  • For oddness, also check whether -f(x) equals f(-x).
• Examples from the lecture:
  • f(x) = x^2 + 4
  • f(-x) = (-x)^2 + 4 = x^2 + 4 = f(x) → even.
  • f(x) = -x^2 - x + 2
  • f(-x) = -(-x)^2 - (-x) + 2 = -x^2 + x + 2
  • This is not equal to f(x) and not equal to -f(x) either → neither even nor odd.
• Quick practice from lecture: test if a given function is even, odd, or neither by computing f(-x) and comparing to f(x) and to -f(x).

Practice problem discussion (blue book style)

• Problem: transform a given parent graph and then draw the result for a specific composite transformation (e.g., from the transcript: minus x plus two, i.e., g(x) = -x + 2 or a variant with inside/outside shifts). The instructor walks through identifying:
  • The base shape (e.g., y = x^2 for a parabola).
  • The horizontal and vertical shifts (left/right and up/down).
  • Any reflections across axes due to negative multipliers.
• The logic used in the discussion:
  • If the problem is something like f(x) = -x + 2, interpret as a line with slope -1 translated so that the line crosses the y-axis at 2.
  • The graph is then reflected across one axis depending on the sign of the multiplier and shifted according to the constants.
• Student interaction notes:
  • A student asked for clarification on how to apply the negative sign when constructing the table for a transformed inside expression (e.g., f(-x) with a negative x). The instructor reminded that the negative sign on x is part of the input substitution; for example, to evaluate f(-2), plug x = -2 into the function, which yields the y-value on the transformed graph.
  • Another student questioned how to place points when the inside transformation moves the vertex (e.g., for a parabola). The instructor stated to shift vertex by the same horizontal and vertical amounts indicated by the outside/inside terms.
• Practical takeaway: when dealing with f(x) + a, f(x - a), f(x + a), f(bx), and -f(x), keep track of each operation's geometric effect (vertical shift, horizontal shift, stretch/compression, and reflection) and verify with an x-y table or by tracking notable points on the graph.

Summary of key ideas to memorize for the exam

• Piecewise functions:
  • Understand how to read the conditions and determine which piece applies at a given x.
  • Be able to determine end behavior at breakpoints and draw open/closed circles appropriately.
• Graphing strategies for piecewise functions:
  • Graph each piece separately and then combine, or evaluate boundary points to locate connecting points.
• Transformation rules (six core rules, plus two extra for scaling):
  • Vertical shifts: f(x) ± k → up/down by k.
  • Horizontal shifts: f(x ± k) → left/right by k.
  • Reflections: -f(x) (across x-axis); f(-x) (across y-axis).
  • Vertical stretch/compression: a f(x) with a > 1 or 0 < a < 1; a < 0 includes reflection.
  • Horizontal stretch/compression: f(bx) with b > 1 (compression), 0 < b < 1 (stretch).
• Inside vs outside intuition: when inside the function argument is altered (x → x ± k or bx), imagine moving the graph; when outside the function, imagine moving the graph up/down or flipping it.
• Even vs odd functions: use f(-x) to classify; even if f(-x) = f(x); odd if f(-x) = -f(x);
• Be prepared to discuss and correct common misconceptions (e.g., misinterpreting the effect of f(-x) on certain odd functions like x^3), and know that f(-x) always yields a reflection about the y-axis, with the exact y-values determined by the original function.
• Practice problems will involve applying multiple transformations in sequence, and it is helpful to sketch both the parent function and the transformed version or check key coordinates with an x-y table.

Notation and formulas to remember

• Piecewise example:
  $f(x)=\begin{cases}x,&amp; x<2\ x^2,& 2\le x\le 4\ 3x+5,& x>4\end{cases}$
• Horizontal shift intuition: f(x+k) shifts left by k; f(x-k) shifts right by k.
• Vertical shift intuition: f(x) + k shifts up by k; f(x) - k shifts down by k.
• Reflections:
  • Across y-axis: f(-x)
  • Across x-axis: -f(x)
• Scaling:
  • Vertical: a f(x) with a > 1 (stretch), 0 < a < 1 (compression), a < 0 (reflection + stretch).
  • Horizontal: f(bx) with b > 1 (compression by 1/b), 0 < b < 1 (stretch by 1/b).
• Even/odd tests:
  • Even: f(-x) = f(x)
  • Odd: f(-x) = -f(x)

Final note

• The lecture blends conceptual explanations with worked examples and student discussions. The essential takeaways are the definitions, the six core transformation rules (plus two scaling rules), and the strategies for graphing piecewise functions and identifying even/odd behavior. Use these as your study anchors for the exam.