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HSF · IF.C.7b

Square Root, Cube Root, and Transformed Root Functions

Deck 1 of 2: Root Function Families

In this deck:

  • Graph the parent square root and cube root functions
  • Apply transformations to root functions using strategic points
Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

What You Will Learn in Deck One

In this deck you will:

  1. Graph y = √x and y = ∛x, identifying domain, range, intercept, and end behavior
  2. Graph transformed square root and cube root functions using strategic points

Build on: domain restrictions (HSF.IF.B.5) and general graphing strategies (HSF.IF.C.7).

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Square Root Parent Function: Domain and Table

0 1 4 9 16
0 1 2 3 4

Use perfect squares — clean integer outputs. Outputs grow slower as inputs grow.

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Graphing f(x) = √x: Shape and Features

Square root parent function with five points plotted and curve drawn; domain starts at origin

  • Domain: ; Range: ; Starting point:
  • Always increasing, concave down; as
Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Cube Root Parent Function: Domain and Table

−8 −1 0 1 8
−2 −1 0 1 2

Use perfect cubes. Extends in both directions — unlike .

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Graphing g(x) = ∛x: S-Curve and Features

  • Domain: ; Range:
  • Inflection point: — concavity changes here
  • Shape: S-curve; concave up for , concave down for

End behavior: as , .

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Comparing √x and ∛x: Key Differences

Side-by-side comparison of square root and cube root parent functions with key features labeled

Choose strategic inputs: perfect squares for √x, perfect cubes for ∛x.

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Check: Domain and Value of Root Functions

  1. What is the domain of ?
  2. Does exist? If so, what is it?
  3. Which root function has a restricted domain?

Answer all three before the next slide.

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Answer: Domain and Root Function Values

  1. Domain of : — requires
  2. — exists, since
  3. has a restricted domain; does not

Key distinction: square root domain starts at zero; cube root domain is all reals.

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Transforming Root Functions: a√(x − h) + k

  • Starting point: ; stretch: ; reflection:
  • ⚠️ Sign trap: starts at , not
Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Transformed Square Root: 2√(x − 3) − 1

Transformed square root function starting at (3,−1) with points (4,1) and (7,3) marked

  • Starting point: ; Domain:
  • Point check: → ; →
Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Reflection: When a Is Negative

  • Reflects the graph over the x-axis
  • Domain: still — reflection doesn't change domain
  • Range: now — all outputs are non-positive

Combine with shifts: reflects, shifts right 2, up 3.

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Transformed Cube Root: ∛(x + 2) − 4

  • Inflection point: ; domain: all reals
  • ; — strategic points for the sketch

S-curve centered at the inflection point .

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Check: Transformed Square Root Features

For :

  1. What is the starting point?
  2. What is the domain?
  3. What is the stretch factor?

Answer all three before the next slide.

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Answer: Features of 3√(x + 5) − 2

  • Starting point: ; domain: ; stretch: 3×
  • Next point: →
Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Key Takeaways from Deck One

✓ : domain , starts at origin, concave down

✓ : domain all reals, S-curve, inflection at origin

✓ Transformation: starting/inflection = ; use perfect squares/cubes

⚠️ is restricted; is not — not interchangeable

⚠️ starts at , not

Graph Root and Piecewise Functions · Slide {page}
HSF · IF.C.7b

Coming Up in Deck Two

Piecewise-Defined Functions, Step Functions, and Absolute Value

  • Graph multi-rule functions with open and closed dots
  • Understand jump discontinuities versus continuous transitions
  • Graph step functions: staircase pattern with correct endpoints
  • Graph absolute value as a piecewise-linear V-shape

Covers LOs 3, 4, 5, and 6

Graph Root and Piecewise Functions · Slide {page}

Click to begin the narrated lesson

Graph root and piecewise functions