Parent Functions & Transformations – Study Notes

Essential Question & Standards

• Essential Question: “What are the characteristics of some of the basic parent functions?”

• Common Core Learning Standard: $HSF\text{-}BF.B.3$ – Identify the effect of transformations on graphs of functions.

Core Vocabulary

• Parent function: Most basic form of a function within a family (no transformations applied).

• Transformation: Any change in size, shape, position, or orientation of a graph.

• Translation: Horizontal and/or vertical shift; size & shape unchanged.

• Reflection: Flip of a graph across a line (usually $x$– or $y$–axis).

• Vertical Stretch: Multiply every $y$–coordinate by a factor k>1 (graph pulls away from $x$–axis).

• Vertical Shrink: Multiply every $y$–coordinate by a factor 0<k<1 (graph pushes toward $x$–axis).

Eight Basic Parent Functions – Key Characteristics

• Constant: $f(x)=1$
  • Graph: horizontal line.
  • Domain: all real numbers.
  • Range: $y=1$.

• Linear: $f(x)=x$
  • Graph: line through origin with slope $1$.
  • Domain/Range: all real numbers.

• Absolute Value: $f(x)=|x|$
  • Graph: V-shape opening upward.
  • Domain: all real numbers.
  • Range: $y\ge0$.

• Quadratic: $f(x)=x^2$
  • Graph: parabola opening upward.
  • Domain: all real numbers.
  • Range: $y\ge0$.

• Square Root: $f(x)=\sqrt{x}$
  • Graph: starts at $(0,0)$, increases slowly.
  • Domain: $x\ge0$.
  • Range: $y\ge0$.

• Cubic: $f(x)=x^3$
  • Graph: S-shaped; passes through origin; odd symmetry.
  • Domain/Range: all real numbers.

• Reciprocal: $f(x)=\frac1x$
  • Graph: two hyperbolic branches in quadrants I & III; asymptotes at $x=0$, $y=0$.
  • Domain/Range: $x\ne0$, $y\ne0$.

• Exponential: $f(x)=2^x$ (generic base b>0,\, b\ne1)
  • Graph: passes through $(0,1)$; rapid growth; horizontal asymptote $y=0$.
  • Domain: all real numbers.
  • Range: y>0.

Identifying Function Families

• Functions in the same family are simply transformations of a single parent.

• Strategy: Inspect the variable term; e.g., if the only variable term is inside $|\ |$, function is absolute value.

Example ➔ Absolute-Value Family

• Given $f(x)=2|x|+1$.
  • Shape: V-shaped ⇒ absolute value.
  • Transformations vs. parent $|x|$:
  – Vertical translation up 1.
  – Vertical stretch by factor 2.
  • Domain stays $\mathbb{R}$, but Range becomes $y\ge1$.

Translation – Shifting Graphs

• Rule: $y=f(x) \to y=f(x-h)+k$
  • h>0 ⇒ shift right $h$.
  • h<0 ⇒ shift left $|h|$. • k>0 ⇒ shift up $k$.
  • k<0 ⇒ shift down $|k|$.

Linear Translation Example

• Parent: $f(x)=x$.

• Transformed: $g(x)=x-4$.
  • Graph: same slope $1$, $y$-intercept at $-4$.
  • Description: 4-unit downward vertical translation.

Reflection – Flipping Graphs

• Across $x$–axis: $y=f(x) \to y=-f(x)$.

• Across $y$–axis: $y=f(x) \to y=f(-x)$.

Quadratic Reflection Example

• Parent: $f(x)=x^2$.

• Transformed: $p(x)=-x^2$.
  • Every $y$ becomes its negative ⇒ parabola opens downward.
  • Described as a reflection across the $x$–axis.

Vertical Stretch & Shrink

• General: $y=f(x) \to y=k\,f(x)$.
  • k>1 ⇒ stretch (graph taller/narrower).
  • 0<k<1 ⇒ shrink (graph shorter/wider).

Absolute-Value Stretch Example

• $g(x)=2|x|$
  • Each $y$ doubled ⇒ vertical stretch by 2.

Quadratic Shrink Example

• $h(x)=\tfrac12 x^2$
  • Each $y$ halved ⇒ vertical shrink by \tfrac12.

Combinations of Transformations

• Transformations can be chained; order often read inside → outside or described stepwise.

Multi-Step Example

• $g(x)=-|x+5|-3$

  1. Inside $x+5$ ⇒ shift left 5.

  2. Outside negative ⇒ reflection across $x$–axis.

  3. Final “−3” ⇒ shift down 3.

• Graph: upside-down V, vertex at $(-5,-3)$ .

Modeling With Mathematics

Dirt-Bike Jump Data

• Scatter plot resembles a symmetric curve ⇒ quadratic model.

• Estimation: At $x=1.75$ s, height $\approx15$ ft. (Consistent with values between 1.5 s & 2 s.)

Graphing-Calculator Skills & Practice Highlights

• Tasks include:
  • Graphing $g(x)=x+3$, $h(x)=(x-2)^2$, $n(x)=-|x|$, etc.
  • Determining translations, reflections, stretches/shrinks.
  • Error analysis: Verify correct description of transformation.
  • Modeling scenarios (temperature change, car depreciation, chainsaw fuel, car speed approaching a stop sign).

• 4-Step Problem-Solving Framework: 1) Understand, 2) Plan, 3) Solve, 4) Look Back.

Helpful Visual & Conceptual Tips

• Visualizing Stretch/Shrink: Imagine pulling points away from or pushing them toward the $x$–axis.

• Slope-Intercept Reminder: For linear $y=mx+b$, $m$ is slope, $b$ $=$ $y$-intercept.

• Ordered-Pair Check: To test if $(a,b)$ lies on $f$, substitute: $f(a)=b$?

• Intercepts: $x$–intercept ⇒ set $y=0$; $y$–intercept ⇒ set $x=0$.

Connections to Earlier Topics

• Domain / Range review (previously studied).

• Scatter plots used for visual pattern recognition.

• Slope concepts reappear in the context of linear parent.

Ethical / Practical Implications Discussed

• None explicitly; focus is procedural fluency & graph interpretation.

Numerical & Algebraic References Captured

• Temperature model: starts $43^\circ!F$ at 8 a.m.; increases $2^\circ!F$ each hour for 7 h (linear).

• Car value: $f(x)=10{,}000-250x^2$ (quadratic depreciation).

• Basketball shot: $f(t)=-16t^2+32t+5.2$ (vertical motion under gravity).

• Chainsaw fuel data: $y$ decreases linearly from 15 oz to 0 oz; empty at $x=50$ min.

Quick Reference – Identifying Transformation Types

Use these templates to decode any function’s graph quickly.