Graphs of Basic Functions and Transformations - Study Notes

Continuity and Discontinuity

• Definition of continuity (intuitive): A function is continuous on an interval if you can draw its graph without lifting your pencil from the paper. In other words, no breaks, holes, or jumps on the graph over the interval.
• Graphical indicator of continuity: A graph that can be drawn in one stroke (without lifting the pen) over the interval.
• Example of discontinuity (at x = 2): A function where the graph has a break at x = 2. The transcript describes a function that is not defined at x = 2 on one side but has a defined value at x = 2 from another piece, creating a discontinuity.
  • Visual cue: an open circle at (2, yleft) and a closed dot at (2, yright). The function value at x = 2 is the closed dot value (y_right).
• Open vs closed dots (key notation):
  • An open dot at a point means the function is not taking that value there (not defined at that point for that branch).
  • A closed dot at a point means the function value at that x is equal to that y-value (included in the function).
• Example of a continuous function (described in the transcript): A function with arrows indicating the graph continues indefinitely in both directions (domain: \(-\infty, \infty\)\) and range starting from 0 upwards (\[0, \infty)\).
• Example of a discontinuous function due to a missing point: A function not defined at x = 3 (domain excludes x = 3) but defined elsewhere; this creates a break in the graph at x = 3.

Describing intervals of continuity for given functions

• Function with unbroken arrows (continuous):
  • Domain: \(-\infty, \infty\)
  • Range: \([0, \infty)\)
• Function with a discontinuity at x = 3:
  • Not defined at x = 3; defined elsewhere.
  • Domain: all real numbers except x = 3, i.e., \((-\infty, 3) \cup (3, \infty)\).

Piecewise-defined functions: graphing and domain/range

• Example 1 (piecewise):
  • Left piece: $f(x) = -2x + 5 ext{ for } x \,\le\, 2$
  • Right piece: f(x) = x + 1 ext{ for } x \,>\, 2
  • At x = 2: left piece would give $f(2) = -2(2) + 5 = 1$, but the transcript shows the left branch has an open circle at (2,1) and the right branch has a closed dot at (2,3). Hence, the actual value is set by the right piece: $f(2) = 3$.
  • Domain (overall): $(-\infty, 2) \cup [2, \infty)$
  • Left-hand limit as x approaches 2 from the left: $\lim_{x\to 2^-} f(x) = 1$ (not included since x < 2 for that branch).
  • Right-hand limit as x approaches 2 from the right: $\lim_{x\to 2^+} f(x) = 3$ (included for x > 2).
  • Range: From the left piece, values approach 1 from above; from the right piece, values start at 3 and go to infinity. Therefore, the combined range is $(1, \infty) ).</li> <li>This function is discontinuous at x = 2 (jump discontinuity).</li></ul></li> <li>Example 2 (another piecewise):<ul> <li>Left piece:$f(x) = 2x + 3 ext{ for } x \le 0$</li> <li>Right piece:$f(x) = -x^2 + 3 ext{ for } x > 0$</li> <li>Discontinuity at x = 0 because the right piece is not defined at x = 0, while the left piece is, and the two pieces do not agree at x = 0.</li> <li>Domain:$(-\infty, 0] \cup (0, \infty)$</li> <li>Range: not evaluated explicitly in the transcript; the key point is the discontinuity at x = 0.</li></ul></li> </ul> <h3 id="importantbasicfunctionstoknow">Important basic functions to know</h3> <ul> <li><p>Identity function:$f(x) = x
    • Domain: \(-\infty, \infty\)
    • Range: \(-\infty, \infty\)
    • Graph: a straight line with slope 1 through the origin.

  • Squaring function: f(x) = x^2$</p> <ul> <li>Graph: a parabola opening upwards with vertex at (0,0).</li></ul></li> <li><p>Cubing function:$f(x) = x^3$</p> <ul> <li>Graph: S-shaped curve passing through the origin; odd symmetry about the origin.</li></ul></li> <li><p>Square root function:$f(x) = \sqrt{x}$</p> <ul> <li>Defined for$x \ge 0$only; graph starts at (0,0) and increases to the right.</li></ul></li> <li><p>Cube root function:$f(x) = \sqrt[3]{x}$</p> <ul> <li>Defined for all real x; passes through origin with a gentle S-curve; no domain restriction.</li></ul></li> <li><p>Absolute value:$f(x) = |x|$</p> <ul> <li>Graph is a V-shaped figure with vertex at the origin; always nonnegative.</li></ul></li> <li><p>Quick note on how these basic shapes help with transformations and recognizing the effect of shifts/scaling.</p></li> <li><p>Additional intuition: knowing these basic functions helps you predict the effect of transformations (translations, stretches/compressions, reflections) on more complicated functions.</p></li> </ul> <h3 id="transformationsofgraphsoverview">Transformations of graphs: overview</h3> <ul> <li>What is a transformation? Changing a function’s graph by stretching/compressing, shifting, reflecting, or rotating.</li> <li>Typical transformations include:<ul> <li>Vertical stretches/compressions</li> <li>Horizontal stretches/compressions</li> <li>Vertical reflections</li> <li>Horizontal reflections</li> <li>Translations (shifts) up/down and left/right</li> <li>Rotations (not covered in detail here)</li></ul></li> <li>The focus here is on translations, stretches/compressions, and reflections.</li> </ul> <h3 id="verticaltransformations">Vertical transformations</h3> <ul> <li>Vertical stretch/compression: If$g(x) = a \, f(x)$, with a a constant:<ul> <li>If$a > 1$: vertical stretch by a factor of$a$.</li> <li>If$0 < a < 1$: vertical compression by a factor of$a$.</li> <li>If$a < 0$: vertical reflection about the x-axis (and possibly a stretch/compression as well, depending on |a|).</li> <li>Example: comparing$f(x) = |x|$and$g(x) = 2|x|$shows a vertical stretch by 2.</li></ul></li> <li>Vertical shift (translation): if$f(x)$is the base function, then$f(x) + k$translates the graph upward by$k$; if$f(x) - k$translates downward by$k$.</li> <li>Note on combining vertical transformation order: multiply (stretch/compress) first, then add/subtract to shift. The arithmetic order is important: scaling before shifting.</li> </ul> <h3 id="horizontaltransformations">Horizontal transformations</h3> <ul> <li>Horizontal stretch/compression:$g(x) = f(bx)$, with b a constant:<ul> <li>If$b > 1$: horizontal compression by a factor of$1/b$.</li> <li>If$0 < b < 1$: horizontal stretch by a factor of$1/b$.</li> <li>If$b < 0$: horizontal reflection about the y-axis (and a combination of stretch/compression depending on |b|).</li></ul></li> <li>Intuition: replacing x by a scaled x compresses or stretches the graph horizontally.</li> <li>Horizontal shift (translation) via the inside of the function: replace x by x − h or x by x + h.<ul> <li>If we have f(x − h): shift right by h.</li> <li>If we have f(x + h): shift left by h.</li></ul></li> <li>The domain of inner transformations is important; the order in which you apply horizontal shifts and horizontal stretches impacts the final graph, especially when both inside the argument are modified (e.g., f(bx + c)).</li> </ul> <h3 id="reflections">Reflections</h3> <ul> <li>Reflection across the x-axis (vertical reflection):$g(x) = -f(x)$<ul> <li>The graph is flipped over the x-axis.</li></ul></li> <li>Reflection across the y-axis (horizontal reflection):$g(x) = f(-x)$<ul> <li>The graph is flipped over the y-axis.</li></ul></li> <li>Practical takeaway: To reflect across axes, multiply by -1 (x-axis) or replace x with -x (y-axis).</li> </ul> <h3 id="workedexampleverticallyandhorizontallytransformedfunctions">Worked example: vertically and horizontally transformed functions</h3> <ul> <li>Example: g(x) = 2|x| is a vertical stretch of the base function$|x| by a factor of 2 (compare the blue and red graphs).

  • Example: h(x) = \tfrac{1}{2} |x| is a vertical compression by a factor of 2 (|a| < 1).
  • Understanding via a table: you can still generate points for the transformed function by applying the transformation rules to the basic function’s points.
  • Important observation: the basic shapes (abs, x^2, sqrt, etc.) help you anticipate how the transformed graph will look.

  Reflections and symmetry (even and odd functions)

  • Even functions: f(x) is even if for all x, f(x) = f(-x). Graphs are symmetric about the y-axis (horizontal reflection).
  • Odd functions: f(x) is odd if for all x, f(x) = -f(-x). Graphs are symmetric about the origin.
  • Quick test approach:
    • Check if f(-x) equals f(x) for all x to test for evenness.
    • Check if f(-x) equals -f(x) for all x to test for oddness.
  • Practical example from the transcript:
    • Consider f(x) = x^3 - 2x. Then f(-x) = -x^3 + 2x = -(x^3 - 2x) = -f(x); hence f is odd.
  • Another example discussed: a polynomial with even powers (like s^4 + 3s^2 + 7 using s as the variable) is even (f(-s) = f(s)).
  • It’s possible to have a function that is neither even nor odd (e.g., f(x) = x^4 + 3x^3 - 7 may fail both tests).

  Combining transformations (order matters when inside the function is transformed)

  • General guidance for combining vertical transformations (outside the function):
    • Apply the vertical stretch/compression first (multiply by a), then apply vertical shift (add k).
    • Example form: g(x) = a f(x) + k$with a ≠ 0.</li></ul></li> <li>General guidance for combining horizontal transformations (inside the function):<ul> <li>When you have an inside transformation like$f(bx + h)$, you typically think of two steps: shift and then scale.</li> <li>A common rule is to rewrite inside as a product:$bx + h = b\left(x + \frac{h}{b}\right)$and interpret as a horizontal shift by$-\frac{h}{b}$followed by a horizontal stretch/compression by factor$1/b$.</li> <li>If you have multiple inside changes, the order is important; however, horizontal and vertical transformations are often viewed as independent in terms of effect on shape, so in some cases the overall order can be considered separately for horizontal vs vertical.</li></ul></li> <li>Summary: For a combined function of the form$f(bx + h)$, you typically consider the horizontal shift by$-\frac{h}{b}$first (if you decompose into a shift then a scale), then apply the horizontal scaling by a factor of$1/b$. The exact geometric order can be clarified by rewriting the inside as$f\left(b\left(x + \frac{h}{b}\right)\right) = f\left(bx + h\right).$</li> <li>Important note: Horizontal and vertical transformations are independent in the sense that one does not constrain the other; you can analyze them separately.</li> </ul> <h3 id="aworkedcompositeexample">A worked composite example</h3> <ul> <li>Original function is a semicircle-ish shape (described as f(x) with a half-circle shape).</li> <li>Consider the transformation:$h(x) = f\left(\tfrac{1}{2}x + 1\right) - 3.$<ul> <li>Step 1 (inside): Rewrite inside as (\tfrac{1}{2}x + 1 = \tfrac{1}{2}(x + 2)).</li> <li>This indicates a horizontal stretch by a factor of 2 (since the coefficient of x is 1/2) and a horizontal shift left by 2 (due to the +2 inside the parentheses after factoring the 1/2).</li> <li>Step 2 (outside): Subtract 3 shifts the entire graph downward by 3 units.</li> <li>Net effect: A horizontal stretch by 2, followed by a left shift by 2, followed by a downward shift by 3.</li></ul></li> <li>Important practical takeaway: when dealing with complex inside/outside transformations, rewrite the inside to understand the order of horizontal changes, then apply any vertical changes afterwards.</li> </ul> <h3 id="practicaltipsandreflections">Practical tips and reflections</h3> <ul> <li>Always connect back to the basic shapes when predicting the transformed graph.</li> <li>When graphing by hand, it can be helpful to use a table of values for the base function and apply the transformation rules to obtain the transformed table, then plot.</li> <li>Remember domain constraints for certain basic functions (e.g., sqrt requires nonnegative input):<ul> <li>For$f(x) = \sqrt{x}\,$, the domain is$x \ge 0.$</li> <li>For$f(x) = \sqrt{-x}\,$, the domain is$x \le 0.$</li></ul></li> <li>Practical domains/ranges and continuity: use them to anticipate where a function exists and how the graph behaves near potential discontinuities.</li> </ul> <h3 id="summaryconnectionstobroadertopics">Summary connections to broader topics</h3> <ul> <li>Continuity ties into the more general study of limits and the behavior of graphs at points of interest (where the function is not defined or where the slope changes abruptly).</li> <li>Piecewise functions introduce the idea that a single equation may not describe the entire domain; you must consider each piece and its domain.</li> <li>Transformations connect to the idea that many graphs are built from simple, well-understood functions; by applying a small set of operations you can model a wide variety of curves.</li> <li>Even and odd functions highlight symmetry properties that simplify analysis and graphing, and they link to fundamental concepts in calculus and symmetry.</li> </ul> <h3 id="quickreferenceformulaslatex">Quick reference formulas (LaTeX)</h3> <ul> <li>Vertical stretch/compression:$g(x) = a\,f(x)$</li> <li>Horizontal stretch/compression:$g(x) = f(bx)$</li> <li>Horizontal shift:$f(x - h)$shifts right by$h$;$f(x + h)$shifts left by$h$</li> <li>Vertical shift:$f(x) + k$shifts up by$k$;$f(x) - k$shifts down by$k$</li> <li>Reflections: across x-axis$g(x) = -f(x)$; across y-axis$g(x) = f(-x)$</li> <li>Combination example:$h(x) = f\left(\tfrac{1}{2}x + 1\right) - 3$</li> <li>Domain and range examples from piecewise cases are described in context above (e.g., domains like$(-\infty, 2) \cup [2, \infty)$$, etc.)

    Note on the lecture context

    • The notes above faithfully reflect the topics discussed: continuity and discontinuity, piecewise functions, six basic functions to know, transformations (vertical/horizontal shifts, stretches/compressions, reflections), even/odd functions, and combining transformations with a worked example.
    • The content emphasizes developing intuition via basic shapes, then applying transformation rules to build more complex graphs.
    • Practical reminder: some of the specifics (like exact numerical domain/range in every example) depend on the precise function definitions given in exercises; use the rules outlined here to analyze similar problems.

    End of notes