Graphing Radical Functions

Worked solutions · Section 4.3, page 212 · Problems 3-10, 13, 19, 20

Square root rules

1. Parent square root function

\(f(x)=\sqrt{x}\)
Parent = original graph
\(x=0:\ y=\sqrt{0}=0\)
\(x=1:\ y=\sqrt{1}=1\)
\(x=4:\ y=\sqrt{4}=2\)
\(x=9:\ y=\sqrt{9}=3\)

2. Horizontal translation

\(\sqrt{x-h}\quad h\text{ units right}\)
Inside the radical: opposite sign
\(\sqrt{x+h}\quad h\text{ units left}\quad(h>0)\)

3. Vertical translation

\(\sqrt{x}+k\quad k\text{ units up}\)
Outside the radical: same sign
\(\sqrt{x}-k\quad k\text{ units down}\quad(k>0)\)

4. Vertical stretch or shrink

\(y=a\sqrt{x-h}+k\)
Multiply the square root by \(a\), then add \(k\)
  • \(a>1\): vertical stretch by a factor of \(a\)
  • \(0<a<1\): vertical shrink by a factor of \(a\)

5. Starting point

\((x,y)=(h,k)\)
Set the radicand (inside the radical) equal to 0

6. Graphing points

\((x,y)=(h+t,\ a\sqrt{t}+k)\)
\(t\) = radicand; choose \(0,1,4,9\)
  • Set the radicand equal to \(t\)
  • Solve for \(x\)
  • Substitute that \(x\) into the function to find \(y\)

7. Domain and range

\(x\ge h\qquad y\ge k\quad(a>0)\)
Domain = allowed \(x\)-values; range = possible \(y\)-values
  • Square root: radicand \(\ge0\)
  • Positive multiplier: square root part \(\ge0\)
  • Closed starting point; curve continues right and up

Problems 3-8: match the function with its graph.

Problem 3

\(f(x)=\sqrt{x+3}\)

Rules used:
  • 2. Horizontal translation
  • 5. Starting point
  • 6. Graphing points
Step 1. Use radicand 0; find the starting point.
\(\begin{gathered}x+3=0\;\Longrightarrow\;x=0-3=-3\\y=\sqrt{-3+3}=0\end{gathered}\)
Step 2. Use radicand 1; find a second point.
\(\begin{gathered}x+3=1\;\Longrightarrow\;x=1-3=-2\\y=\sqrt{-2+3}=1\end{gathered}\)
Step 3. Match both points.
\(x\)\(-3\)\(-2\)
\(y\)\(0\)\(1\)
Graph B
3 units left
Problem 4

\(h(x)=\sqrt{x}+3\)

Rules used:
  • 3. Vertical translation
  • 5. Starting point
  • 6. Graphing points
Step 1. Use radicand 0; find the starting point.
\(\begin{gathered}x=0\\y=\sqrt{0}+3=3\end{gathered}\)
Step 2. Use radicand 1; find a second point.
\(\begin{gathered}x=1\\y=\sqrt{1}+3=4\end{gathered}\)
Step 3. Match both points.
\(x\)\(0\)\(1\)
\(y\)\(3\)\(4\)
Graph D
3 units up
Problem 5

\(f(x)=\sqrt{x-3}\)

Rules used:
  • 2. Horizontal translation
  • 5. Starting point
  • 6. Graphing points
Step 1. Use radicand 0; find the starting point.
\(\begin{gathered}x-3=0\;\Longrightarrow\;x=0+3=3\\y=\sqrt{3-3}=0\end{gathered}\)
Step 2. Use radicand 1; find a second point.
\(\begin{gathered}x-3=1\;\Longrightarrow\;x=1+3=4\\y=\sqrt{4-3}=1\end{gathered}\)
Step 3. Match both points.
\(x\)\(3\)\(4\)
\(y\)\(0\)\(1\)
Graph F
3 units right
Problem 6

\(g(x)=\sqrt{x}-3\)

Rules used:
  • 3. Vertical translation
  • 5. Starting point
  • 6. Graphing points
Step 1. Use radicand 0; find the starting point.
\(\begin{gathered}x=0\\y=\sqrt{0}-3=-3\end{gathered}\)
Step 2. Use radicand 1; find a second point.
\(\begin{gathered}x=1\\y=\sqrt{1}-3=-2\end{gathered}\)
Step 3. Match both points.
\(x\)\(0\)\(1\)
\(y\)\(-3\)\(-2\)
Graph A
3 units down
Problem 7

\(h(x)=\sqrt{x+3}-3\)

Rules used:
  • 2. Horizontal translation
  • 3. Vertical translation
  • 5. Starting point
  • 6. Graphing points
Step 1. Use radicand 0; find the starting point.
\(\begin{gathered}x+3=0\;\Longrightarrow\;x=0-3=-3\\y=\sqrt{-3+3}-3=-3\end{gathered}\)
Step 2. Use radicand 1; find a second point.
\(\begin{gathered}x+3=1\;\Longrightarrow\;x=1-3=-2\\y=\sqrt{-2+3}-3=-2\end{gathered}\)
Step 3. Match both points.
\(x\)\(-3\)\(-2\)
\(y\)\(-3\)\(-2\)
Graph E
3 units left, 3 units down
Problem 8

\(f(x)=\sqrt{x-3}+3\)

Rules used:
  • 2. Horizontal translation
  • 3. Vertical translation
  • 5. Starting point
  • 6. Graphing points
Step 1. Use radicand 0; find the starting point.
\(\begin{gathered}x-3=0\;\Longrightarrow\;x=0+3=3\\y=\sqrt{3-3}+3=3\end{gathered}\)
Step 2. Use radicand 1; find a second point.
\(\begin{gathered}x-3=1\;\Longrightarrow\;x=1+3=4\\y=\sqrt{4-3}+3=4\end{gathered}\)
Step 3. Match both points.
\(x\)\(3\)\(4\)
\(y\)\(3\)\(4\)
Graph C
3 units right, 3 units up

Problems 9, 10, 13: graph the function.

Identify the domain and range of the function.

Problem 9

\(h(x)=\sqrt{x}+4\)

Rules used:
  • 3. Vertical translation
  • 5. Starting point
  • 6. Graphing points
  • 7. Domain and range

4 units up

Step 1. Use radicand 0; solve for \(x\), then find \(y\).
\(\begin{gathered}x=0\\y=\sqrt{0}+4=4\end{gathered}\)
Step 2. Use radicand 1; solve for \(x\), then find \(y\).
\(\begin{gathered}x=1\\y=\sqrt{1}+4=5\end{gathered}\)
Step 3. Use radicand 4; solve for \(x\), then find \(y\).
\(\begin{gathered}x=4\\y=\sqrt{4}+4=6\end{gathered}\)
Step 4. Use radicand 9; solve for \(x\), then find \(y\).
\(\begin{gathered}x=9\\y=\sqrt{9}+4=7\end{gathered}\)
Step 5. Plot the points; draw a smooth curve.
\(x\)\(0\)\(1\)\(4\)\(9\)
\(y\)\(4\)\(5\)\(6\)\(7\)
Problem 9: square root graphSolid green curve with closed starting point (0, 4), continuing right and upward.024681002468xy
Step 6. Find the domain and range.
\(x\ge0\)
\(\sqrt{x}\ge0\quad\Longrightarrow\quad y\ge4\)
Domain and range
Domain: \(x\ge0\)
Range: \(y\ge4\)
Problem 10

\(g(x)=\sqrt{x}-5\)

Rules used:
  • 3. Vertical translation
  • 5. Starting point
  • 6. Graphing points
  • 7. Domain and range

5 units down

Step 1. Use radicand 0; solve for \(x\), then find \(y\).
\(\begin{gathered}x=0\\y=\sqrt{0}-5=-5\end{gathered}\)
Step 2. Use radicand 1; solve for \(x\), then find \(y\).
\(\begin{gathered}x=1\\y=\sqrt{1}-5=-4\end{gathered}\)
Step 3. Use radicand 4; solve for \(x\), then find \(y\).
\(\begin{gathered}x=4\\y=\sqrt{4}-5=-3\end{gathered}\)
Step 4. Use radicand 9; solve for \(x\), then find \(y\).
\(\begin{gathered}x=9\\y=\sqrt{9}-5=-2\end{gathered}\)
Step 5. Plot the points; draw a smooth curve.
\(x\)\(0\)\(1\)\(4\)\(9\)
\(y\)\(-5\)\(-4\)\(-3\)\(-2\)
Problem 10: square root graphSolid green curve with closed starting point (0, -5), continuing right and upward.0246810-5-4-3-2-101xy
Step 6. Find the domain and range.
\(x\ge0\)
\(\sqrt{x}\ge0\quad\Longrightarrow\quad y\ge-5\)
Domain and range
Domain: \(x\ge0\)
Range: \(y\ge-5\)
Problem 13

\(g(x)=\dfrac15\sqrt{x-3}\)

Rules used:
  • 2. Horizontal translation
  • 4. Vertical stretch or shrink
  • 5. Starting point
  • 6. Graphing points
  • 7. Domain and range

3 units right; vertical shrink by a factor of \(\dfrac15\)

Step 1. Use radicand 0; solve for \(x\), then find \(y\).
\(\begin{gathered}x-3=0\;\Longrightarrow\;x=0+3=3\\y=\dfrac{1}{5}\sqrt{3-3}=0\end{gathered}\)
Step 2. Use radicand 1; solve for \(x\), then find \(y\).
\(\begin{gathered}x-3=1\;\Longrightarrow\;x=1+3=4\\y=\dfrac{1}{5}\sqrt{4-3}=\dfrac{1}{5}\end{gathered}\)
Step 3. Use radicand 4; solve for \(x\), then find \(y\).
\(\begin{gathered}x-3=4\;\Longrightarrow\;x=4+3=7\\y=\dfrac{1}{5}\sqrt{7-3}=\dfrac{2}{5}\end{gathered}\)
Step 4. Use radicand 9; solve for \(x\), then find \(y\).
\(\begin{gathered}x-3=9\;\Longrightarrow\;x=9+3=12\\y=\dfrac{1}{5}\sqrt{12-3}=\dfrac{3}{5}\end{gathered}\)
Step 5. Plot the points; draw a smooth curve.
\(x\)\(3\)\(4\)\(7\)\(12\)
\(y\)\(0\)\(\dfrac{1}{5}\)\(\dfrac{2}{5}\)\(\dfrac{3}{5}\)
Problem 13: square root graphSolid green curve with closed starting point (3, 0), continuing right and upward.03691200.20.40.60.81xy
Step 6. Find the domain and range.
\(x-3\ge0\quad\Longrightarrow\quad x\ge3\)
\(\dfrac{1}{5}\sqrt{x-3}\ge0\quad\Longrightarrow\quad y\ge0\)
Domain and range
Domain: \(x\ge3\)
Range: \(y\ge0\)

Problems 19-20: describe the transformation of \(f\) represented by \(g\).

Then graph each function.

Problem 19

\(g(x)=\sqrt{x+1}+8\)

Original function: \(f(x)=\sqrt{x}\)

Rules used:
  • 1. Parent square root function
  • 2. Horizontal translation
  • 3. Vertical translation
  • 5. Starting point
  • 6. Graphing points

1 unit left, 8 units up

Step 1. Use radicand 0; solve for \(x\), then find \(y\).
\(\begin{gathered}x+1=0\;\Longrightarrow\;x=0-1=-1\\y=\sqrt{-1+1}+8=8\end{gathered}\)
Step 2. Use radicand 1; solve for \(x\), then find \(y\).
\(\begin{gathered}x+1=1\;\Longrightarrow\;x=1-1=0\\y=\sqrt{0+1}+8=9\end{gathered}\)
Step 3. Use radicand 4; solve for \(x\), then find \(y\).
\(\begin{gathered}x+1=4\;\Longrightarrow\;x=4-1=3\\y=\sqrt{3+1}+8=10\end{gathered}\)
Step 4. Use radicand 9; solve for \(x\), then find \(y\).
\(\begin{gathered}x+1=9\;\Longrightarrow\;x=9-1=8\\y=\sqrt{8+1}+8=11\end{gathered}\)
Step 5. Plot the parent points and the new points.

Parent \(f\): \(y=\sqrt{x}\)

\(x\)\(0\)\(1\)\(4\)\(9\)
\(y\)\(0\)\(1\)\(2\)\(3\)

New function \(g\)

\(x\)\(-1\)\(0\)\(3\)\(8\)
\(y\)\(8\)\(9\)\(10\)\(11\)
Step 6. Draw both curves from their starting points.
Problem 19: square root graphSolid green curve with closed starting point (-1, 8), continuing right and upward. Dashed gray curve: f(x) = square root of x, starting at (0, 0).-20246810024681012xy

Solid green: \(g\) · Dashed gray: \(f\)

Transformation
1 unit left, 8 units up
Problem 20

\(g(x)=2\sqrt{x-1}\)

Original function: \(f(x)=\sqrt{x}\)

Rules used:
  • 1. Parent square root function
  • 2. Horizontal translation
  • 4. Vertical stretch or shrink
  • 5. Starting point
  • 6. Graphing points

1 unit right; vertical stretch by a factor of 2

Step 1. Use radicand 0; solve for \(x\), then find \(y\).
\(\begin{gathered}x-1=0\;\Longrightarrow\;x=0+1=1\\y=2\sqrt{1-1}=0\end{gathered}\)
Step 2. Use radicand 1; solve for \(x\), then find \(y\).
\(\begin{gathered}x-1=1\;\Longrightarrow\;x=1+1=2\\y=2\sqrt{2-1}=2\end{gathered}\)
Step 3. Use radicand 4; solve for \(x\), then find \(y\).
\(\begin{gathered}x-1=4\;\Longrightarrow\;x=4+1=5\\y=2\sqrt{5-1}=4\end{gathered}\)
Step 4. Use radicand 9; solve for \(x\), then find \(y\).
\(\begin{gathered}x-1=9\;\Longrightarrow\;x=9+1=10\\y=2\sqrt{10-1}=6\end{gathered}\)
Step 5. Plot the parent points and the new points.

Parent \(f\): \(y=\sqrt{x}\)

\(x\)\(0\)\(1\)\(4\)\(9\)
\(y\)\(0\)\(1\)\(2\)\(3\)

New function \(g\)

\(x\)\(1\)\(2\)\(5\)\(10\)
\(y\)\(0\)\(2\)\(4\)\(6\)
Step 6. Draw both curves from their starting points.
Problem 20: square root graphSolid green curve with closed starting point (1, 0), continuing right and upward. Dashed gray curve: f(x) = square root of x, starting at (0, 0).024681001234567xy

Solid green: \(g\) · Dashed gray: \(f\)

Transformation
1 unit right; vertical stretch by a factor of 2

Quick checklist

  1. Inside or outside the radical: horizontal or vertical change
  2. Starting point: radicand \(=0\)
  3. Next points: radicands \(1,4,9\)
  4. Positive multiplier: domain \(x\ge h\), range \(y\ge k\)

Big Ideas Math Integrated Mathematics III (2016), Section 4.3, p. 212 · CCSS HSF-IF.C.7b, HSF-BF.B.3