Graphing Root Functions

Worked solutions · Section 4.3, page 212 · Problems worksheet 1-8

Reading a root function

\(y = a\sqrt[n]{x-h} + k\), and \(n\) = index

1. The four numbers

\(y = a\sqrt[n]{x-h} + k\)
Read \(a\), \(n\), \(h\), \(k\) before anything else

2. a stretches or shrinks

\(|a| > 1:\ \text{stretch}\)
\(0 < |a| < 1:\ \text{shrink}\)
Vertical: every \(y\) value times \(a\)

3. Negative a reflects

\(y = -a\sqrt[n]{x}\)
Reflection over the \(x\)-axis, so the curve turns downward

4. h shifts sideways

\(\sqrt[n]{x-3}:\ 3 \ \text{right}\)
\(\sqrt[n]{x+3}:\ 3 \ \text{left}\)
Opposite of the sign inside

5. k shifts up or down

\(y = \sqrt[n]{x} + k\)
Same sign as written. The starting point of an even root lands at \((h, k)\)

Domain and range

6. Even index

\(D:\ x \ge h\)
\(R:\ y \ge k \ (a > 0), \quad y \le k \ (a < 0)\)
Square, fourth, sixth, eighth. No negative radicand, so the graph starts at \((h,k)\)

7. Odd index

\(D:\ \text{all reals}\)
\(R:\ \text{all reals}\)
Cube, fifth, seventh. Any radicand, so the curve runs both ways

Plotting and writing equations

8. Points without fractions

\(x - h = \text{a perfect } n\text{th power}\)
Pick the radicand first, then read off \(x\). With a fraction out front, pick powers that cancel it

9. Perfect powers worth knowing

\(\text{squares: } 0, 1, 4, 9, 16, 25, 36, 49, 64, 81\)
Pick the radicand from one of these lists
  • Cubes: 0, 1, 8, 27, 64, 125
  • Fifth: 0, 1, 32, 243
  • Sixth: 0, 1, 64, 729
  • Eighth: 0, 1, 256, 6561
  • Odd index: negative radicands too, \(\sqrt[3]{-8} = -2\)

10. Equation from characteristics

\(y = a\sqrt[n]{x-h} + k\)
Shifts and reflection first, then substitute the point and solve for \(a\)

Problems 1-5: Describe transformations of each function. Find domain and range. Sketch the graph (label with points without fractions).

Problem 1

\(f(x) = -2\sqrt[3]{x} - 1\)

Rules used:
Step 1. Read the four numbers.
\(a = -2\), \(n = 3\), \(h = 0\), \(k = -1\)
Step 2. Name the transformations.
Reflection over the \(x\)-axis, vertical stretch by 2, down 1
Step 3. Domain and range.
Odd index, so both are all real numbers
Step 4. Points: cubes for the radicand.
\(f(-8) = -2(-2) - 1 = 3\)
\(f(-1) = -2(-1) - 1 = 1\)
\(f(0) = -1\)
\(f(1) = -2(1) - 1 = -3\)
\(f(8) = -2(2) - 1 = -5\)
Answer
\(D: \text{all reals},\quad R: \text{all reals}\)
Plot (-8, 3), (-1, 1), (0, -1), (1, -3), (8, -5)
Problem 2

\(g(x) = \dfrac{1}{3}\sqrt{x+2} - 3\)

Rules used:
Step 1. Read the four numbers.
\(a = \frac{1}{3}\), \(n = 2\), \(h = -2\), \(k = -3\)
Step 2. Name the transformations.
Vertical shrink by \(\frac{1}{3}\), left 2, down 3
Step 3. Domain and range.
Even index, starting point \((-2, -3)\), opens up
Step 4. Points: squares that divide by 3.
\(g(-2) = -3\)
\(g(7) = \dfrac{1}{3}(3) - 3 = -2\)
\(g(34) = \dfrac{1}{3}(6) - 3 = -1\)
\(g(79) = \dfrac{1}{3}(9) - 3 = 0\)
Answer
\(D: x \ge -2,\quad R: y \ge -3\)
Plot (-2, -3), (7, -2), (34, -1), (79, 0)
Problem 3

\(h(x) = 4\sqrt[5]{x+3} - 2\)

Rules used:
Step 1. Read the four numbers.
\(a = 4\), \(n = 5\), \(h = -3\), \(k = -2\)
Step 2. Name the transformations.
Vertical stretch by 4, left 3, down 2
Step 3. Domain and range.
Odd index, so both are all real numbers
Step 4. Points: fifth powers for the radicand.
\(h(-35) = 4(-2) - 2 = -10\)
\(h(-4) = 4(-1) - 2 = -6\)
\(h(-3) = -2\)
\(h(-2) = 4(1) - 2 = 2\)
\(h(29) = 4(2) - 2 = 6\)
Answer
\(D: \text{all reals},\quad R: \text{all reals}\)
Plot (-35, -10), (-4, -6), (-3, -2), (-2, 2), (29, 6)
Problem 4

\(k(x) = 3\sqrt[6]{x} - 5\)

Rules used:
Step 1. Read the four numbers.
\(a = 3\), \(n = 6\), \(h = 0\), \(k = -5\)
Step 2. Name the transformations.
Vertical stretch by 3, down 5
Step 3. Domain and range.
Even index, starting point \((0, -5)\), opens up
Step 4. Points: sixth powers.
\(k(0) = -5\)
\(k(1) = 3(1) - 5 = -2\)
\(k(64) = 3(2) - 5 = 1\)
\(k(729) = 3(3) - 5 = 4\)
Answer
\(D: x \ge 0,\quad R: y \ge -5\)
Plot (0, -5), (1, -2), (64, 1), (729, 4)
Problem 5

\(h(x) = \dfrac{4}{5}\sqrt[8]{x+3} - 2\)

Rules used:
Step 1. Read the four numbers.
\(a = \frac{4}{5}\), \(n = 8\), \(h = -3\), \(k = -2\)
Step 2. Name the transformations.
Vertical shrink by \(\frac{4}{5}\), left 3, down 2
Step 3. Domain and range.
Even index, starting point \((-3, -2)\), opens up
Step 4. Points: eighth powers.
\(h(-3) = -2\)
\(h(-2) = \dfrac{4}{5}(1) - 2 = -\dfrac{6}{5}\)
\(h(253) = \dfrac{4}{5}(2) - 2 = -\dfrac{2}{5}\)
\(h(6558) = \dfrac{4}{5}(3) - 2 = \dfrac{2}{5}\)
Step 5. Fraction-free points here.
Only \((-3, -2)\); the next clean \(y\) sits past \(x = 390{,}622\)
Answer
\(D: x \ge -3,\quad R: y \ge -2\)
Plot (-3, -2), (-2, -1.2), (253, -0.4), (6558, 0.4)

Problems 6-8: Write an equation of a root function with given characteristics.

Problem 6

\(y = -a\sqrt{x+2} - 1\)

Rules used:
Step 1. Put in the shifts and the reflection.
Reflected, 2 left, 1 down: \(y = -a\sqrt{x+2} - 1\)
Step 2. Substitute the given point.
\((x_1, y_1)\) into the equation, then solve for \(a\)
Step 3. Worked example, point (7, -4).
\(-4 = -3a - 1\)
\(-3 = -3a\)
\(a = 1\)
Step 4. Write it out.
With \(a = 1\): \(y = -\sqrt{x+2} - 1\)
Solution
\(y = -a\sqrt{x+2} - 1\)
Substitute the worksheet's point to find a
Problem 7

\(y = a\sqrt[5]{x-4} \ \text{through}\ (247, 9)\)

Rules used:
Step 1. Put in the shift.
4 right: \(y = a\sqrt[5]{x-4}\)
Step 2. Substitute (247, 9).
\(9 = a\sqrt[5]{243}\)
\(9 = 3a\)
Step 3. Solve for a.
\(a = 3\)
Step 4. Check.
\(3\sqrt[5]{247-4} = 3(3) = 9\)
Solution
\(y = 3\sqrt[5]{x-4}\)
Problem 8

\(y = -a\sqrt[3]{x} - 3\)

Rules used:
Step 1. Put in the reflection and the shift.
Reflected, 3 down: \(y = -a\sqrt[3]{x} - 3\)
Step 2. Substitute the given point.
\((x_1, y_1)\) into the equation, then solve for \(a\)
Step 3. Worked example, point (-64, 1).
\(1 = 4a - 3\)
\(4 = 4a\)
\(a = 1\)
Step 4. Write it out.
With \(a = 1\): \(y = -\sqrt[3]{x} - 3\)
Solution
\(y = -a\sqrt[3]{x} - 3\)
Substitute the worksheet's point to find a

Quick checklist

  1. Four numbers first? \(a\), \(n\), \(h\), \(k\) out of \(y = a\sqrt[n]{x-h}+k\)
  2. Index even or odd? Even starts at \((h,k)\), odd runs both ways
  3. Is \(a\) negative? Reflection over the \(x\)-axis, and the range flips to \(y \le k\)
  4. Sign inside? \(x-h\) moves right, \(x+h\) moves left
  5. Domain of an even root? \(x \ge h\). Odd root: all real numbers
  6. Points without fractions? Make the radicand a perfect \(n\)th power
  7. Writing the equation? Shifts and reflection first, then the point gives \(a\)