Graphing and Transformations of Square Root and Cube Root Functions

Foundations of Quadratic Graphs and Vertex Form

• Quadratic Parent Function:

  • The foundational equation for a quadratic parabola is:     $y = x^2$
  • A balanced table of values includes two negative values, zero, and two positive values:
  • When $x = -2$, $y = (-2)^2 = 4$, yielding the point $(-2, 4)$.
  • When $x = -1$, $y = (-1)^2 = 1$, yielding the point $(-1, 1)$.
  • When $x = 0$, $y = 0^2 = 0$, yielding the origin $(0, 0)$.
  • When $x = 1$, $y = 1^2 = 1$, yielding the point $(1, 1)$.
  • When $x = 2$, $y = 2^2 = 4$, yielding the point $(2, 4)$.
  • Connecting these five coordinate points produces the characteristic U-shaped parabolic curve symmetrical across the vertical axis.

• Vertex Form Structure and Parameters:

  • The general vertex form of a quadratic equation is defined as:     $y = a(x - h)^2 + k$
  • Parameter $h$ governs horizontal translation (shifts left or right):
  • A term of $(x - h)$ indicates a shift to the right by $h$ units.
  • A term of $(x + h)$ indicates a shift to the left by $h$ units.
  • Horizontal movement direction is always opposite to the arithmetic sign inside the parentheses.
  • Parameter $k$ governs vertical translation (shifts up or down):
  • A positive $k$ value shifts the curve upward.
  • A negative $k$ value shifts the curve downward.
  • Parameter $a$ governs vertical dilation (stretching or compressing) and vertical reflection.

• Application to a Shifted Parabola:

  • Consider the quadratic function:     $y = (x - 2)^2 + 3$
  • The equation indicates a horizontal shift of $2$ units to the right and a vertical shift of $3$ units upward.
  • The vertex is located at the ordered pair $(2, 3)$.
  • The vertical axis of symmetry is defined by the line $x = 2$, passing directly through the vertex.
  • Determination of the $y$-intercept occurs by evaluating the function at $x = 0$:     $y = (0 - 2)^2 + 3 = (-2)^2 + 3 = 4 + 3 = 7$
  • This produces the $y$-intercept coordinate $(0, 7)$.
  • Reflecting $(0, 7)$ across the axis of symmetry $x = 2$ generates the corresponding coordinate $(4, 7)$.

The "New Origin" Framework for Function Translations

• Definition and Mechanism:
  • Rather than repeatedly evaluating complex coordinates from the original $(0, 0)$ coordinate origin, a translated graph can be plotted by establishing a "new origin" (or "new home") directly at the shifted vertex $(h, k)$.
  • Dashed lines are placed horizontally and vertically through the coordinates $(h, k)$ to establish a relocated reference frame.
  • From this relocated origin, standard parent function displacement offsets are applied directly:
  • Move $1$ unit right, move $1$ unit up ($1^2 = 1$).
  • Move $2$ units right, move $4$ units up ($2^2 = 4$).
  • Move $1$ unit left, move $1$ unit up ($(-1)^2 = 1$).
  • Move $2$ units left, move $4$ units up ($(-2)^2 = 4$).
  • This procedure constructs an identical parabola without the need to calculate individual function inputs.

The Parent Square Root Function

• Real-Number Domain Restrictions:

  • Evaluating functions within a real-number coordinate axis requires real outputs.
  • Negative values placed under an even radical (square root) yield imaginary/non-real values (for example, $\sqrt{-2}$ is not a real number).
  • Consequently, the domain of the parent square root function is strictly non-negative real numbers:     $\text{Domain: } [0, \infty)$

• Selection of Convenient Table Points:

  • While inputs such as $x = 2$ yield decimal values ($\sqrt{2} \approx 1.4$) and $x = 3$ yield decimal values ($\sqrt{3} \approx 1.6$), integer coordinate points are generated using perfect squares:
  • When $x = 0$, $y = \sqrt{0} = 0$, giving $(0, 0)$.
  • When $x = 1$, $y = \sqrt{1} = 1$, giving $(1, 1)$.
  • When $x = 4$, $y = \sqrt{4} = 2$, giving $(4, 2)$.
  • When $x = 9$, $y = \sqrt{9} = 3$, giving $(9, 3)$.
  • When $x = 16$, $y = \sqrt{16} = 4$, giving $(16, 4)$.

• Geometric Profile and Core Properties:

  • The equation of the parent square root function is:     $y = \sqrt{x}$
  • Geometrically, the curve represents the upper half of a quadratic parabola rotated $90^\circ$, functioning as the inverse of the positive branch of the quadratic parent function.
  • Range Profile:
  • Conceptually described as an elevator starting on Floor $0$ and going upward indefinitely.
  • Although vertical growth progresses at an increasingly slow rate, the curve increases without bound:       $\text{Range: } [0, \infty)$

Transformations of the Square Root Function

• Horizontal and Vertical Translations:

  • Consider the transformed function:     $y = \sqrt{x - 2} + 3$
  • The horizontal shift is $2$ units to the right.
  • The vertical shift is $3$ units upward.
  • The new origin (new home) relocates from $(0, 0)$ to $(2, 3)$.

• Plotting via the New Origin:

  • Rather than evaluating inputs algebraically, parent offset movements are applied relative to the new origin at $(2, 3)$:
  • Move $1$ unit right, $1$ unit up: plotted at $(2 + 1, 3 + 1) = (3, 4)$.
  • Move $4$ units right, $2$ units up: plotted at $(2 + 4, 3 + 2) = (6, 5)$.
  • Move $9$ units right, $3$ units up: plotted at $(2 + 9, 3 + 3) = (11, 6)$.
  • Transformed Domain and Range for this curve:     $\text{Domain: } [2, \infty)$$\text{Range: } [3, \infty)$

Reflections and Vertical Stretches of Radical Functions

• Vertical Reflections:

  • Placing a negative sign in front of the radical creates a vertical reflection across the horizontal axis of the new origin:     $y = -\sqrt{x - 2} + 3$
  • Every standard vertical offset in the parent $y$-table is multiplied by $-1$:
  • Relative offset $(1, 1)$ becomes $(1, -1)$.
  • Relative offset $(4, 2)$ becomes $(4, -2)$.
  • Relative offset $(9, 3)$ becomes $(9, -3)$.
  • From the origin at $(2, 3)$, the plotted points become:
  • $(2 + 1, 3 - 1) = (3, 2)$
  • $(2 + 4, 3 - 2) = (6, 1)$
  • $(2 + 9, 3 - 3) = (11, 0)$
  • The transformed range reflects downward:     $\text{Range: } (-\infty, 3]$

• Vertical Stretches:

  • An absolute value of the leading coefficient $a$ greater than $1$ results in a vertical dilation (stretch) by a factor of $a$.
  • For an equation featuring $a = 2$:
  • All vertical outputs in the $y$-column are doubled ($\times 2$).
  • The relative parent point $(1, 1)$ scales vertically to $(1, 2)$.
  • The relative parent point $(4, 2)$ scales vertically to $(4, 4)$.
  • The relative parent point $(9, 3)$ scales vertically to $(9, 6)$.

The Cube Root Function: Characteristics and Transformations

• Characteristics of the Parent Cube Root:

  • The equation of the parent cube root function is:     $y = \sqrt[3]{x}$
  • Negative inputs under an odd index radical remain defined within the real-number system (e.g., $\sqrt[3]{-8} = -2$ and $\sqrt[3]{-1} = -1$).
  • Standard integer reference coordinates for the parent cube root function include:
  • $(-8, -2)$
  • $(-1, -1)$
  • $(0, 0)$
  • $(1, 1)$
  • $(8, 2)$
  • Domain and Range for the cube root function encompass all real numbers:     $\text{Domain: } (-\infty, \infty)$$\text{Range: } (-\infty, \infty)$
  • Vertical growth and decrease to positive and negative infinity occur at a slow pace, but continue without bound.

• Translation Example:

  • A cube root function with horizontal and vertical shifts such as shifting $1$ unit left and $2$ units down establishes its new home (origin) at:     $(-1, -2)$
  • Plotting parent offsets from the new origin $(-1, -2)$:
  • Left $8$, down $2$: plotted at $(-1 - 8, -2 - 2) = (-9, -4)$.
  • Left $1$, down $1$: plotted at $(-1 - 1, -2 - 1) = (-2, -3)$.
  • Origin point: plotted at $(-1, -2)$.
  • Right $1$, up $1$: plotted at $(-1 + 1, -2 + 1) = (0, -1)$.
  • Right $8$, up $2$: plotted at $(-1 + 8, -2 + 2) = (7, 0)$.

• Reflection of the Cube Root Function:

  • Placing a negative sign in front of the cube root expression inverts all vertical displacements across the new origin's axis:
  • Input offset $-8$ produces output offset $+2$ ($-2 \times -1 = 2$).
  • Input offset $-1$ produces output offset $+1$ ($-1 \times -1 = 1$).
  • Input offset $0$ produces output offset $0$.
  • Input offset $1$ produces output offset $-1$ ($1 \times -1 = -1$).
  • Input offset $8$ produces output offset $-2$ ($2 \times -1 = -2$).
  • Plotting inverted offsets from the new home at $(-1, -2)$:
  • Move left $8$, up $2$: plotted at $(-1 - 8, -2 + 2) = (-9, 0)$.
  • Move left $1$, up $1$: plotted at $(-1 - 1, -2 + 1) = (-2, -1)$.
  • Relocated origin: plotted at $(-1, -2)$.
  • Move right $1$, down $1$: plotted at $(-1 + 1, -2 - 1) = (0, -3)$.
  • Move right $8$, down $2$: plotted at $(-1 + 8, -2 - 2) = (7, -4)$.

Summary of Radical Graph Analysis and Assignments

• Analysis Checklist for Radical Functions:

  • Identify the horizontal translation ($h$) and vertical translation ($k$) to relocate the origin.
  • Establish the new origin at the point $(h, k)$.
  • Identify any vertical dilation factor ($a$) or reflection across the horizontal axis (sign of $a$).
  • Use the standard integer offset table for the corresponding parent function (square root: $0, 1, 4, 9, 16$; cube root: $-8, -1, 0, 1, 8$).
  • Identify and record the domain and range using interval notation.

• Practice Worksheets:

  • Two independent practice sheets ("way grounds") are assigned:
  • Practice set 1: Dedicated to square root graphs, shifts, domains, and ranges.
  • Practice set 2: Dedicated to cube root graphs, shifts, domains, and ranges.

Questions & Discussion

• Classroom Management Interruption:

  • Teacher: "Spot. Is your conversation more important than mine right now?"
  • Student Response: "Sounds good."
  • Teacher: "K."

• Vertex Form Identification Question:

  • Teacher Question: "What moves it left and right? What turn? Move to left and right. In vertex form. I'll put it in vertex form for you. Maybe that'll help. Remember vertex form? What moves it left and right? What's that?"
  • Student Answer: "A."
  • Teacher Correction: "Not the a."
  • Student Answer: "H."
  • Teacher Confirmation: "H. Very good. The h term. The h term moves it left and right. But you remember if it's minus, which way are we going?"
  • Student Answer: "Right."
  • Teacher Confirmation: "To the right. Left is gonna be positive. So it's the opposite of what we see inside. And then the k term moved it up and down. Right?"

• Incidental Classroom Exchanges on Costs and Supplies:

  • Exchange regarding price and bandages:
  • Question/Remark: "$\$8$? How much did you… The bandage on?"
  • Response/Remark: "$48$."
  • Inquiry: "How much were they?"
  • Answer: "$95$ to total."
  • Inquiry: "How old are they? I know it's not really difficult. Just grab something to paint here. It's out there.. Oh. I'm just."
  • Dialogue snippet: "$21$. Oh, squared. One point. K. Hubert squared. Are you doing? You're paying $2$. How much is it gonna be? Oh, it's. Oh, I agree with that. And that was a really good. Really, really happy."