Exponential Skipping Tables: Deriving the Rule and Using Roots

Concept: Skipping Tables and Exponential Progressions

• In a skipping table, the x-values do not increase by 1, but the y-values follow a constant ratio r.
• As you move down the table, you are multiplying by r at each step (geometric progression).
• You don't need to know the value at every intermediate x; you can relate two known values to find the common ratio r.

Deriving the rule from 10 and 22.5

• Suppose the top known value is 10 (at some x0) and the later known value is 22.5 (at x = x0 + 2 steps).
• Since the table is exponential, the relationship between these two values is:
  
  22.5 = 10 \, r^{2}
  
• Solve for r:
  
  r^{2} = \frac{22.5}{10} = 2.25 \
  r = \sqrt{2.25} = 1.5
  
• Important: R is the same regardless of which pair of points you use (as long as you account for the gap in x).
• If you anchor the rule at the top value (x = 3 corresponds to y = 10), the general rule is:
  
  y = 10 \cdot r^{\,x - 3}
  
• With r = 1.5, the equation becomes:
  
  y = 10 \cdot (1.5)^{\,x - 3}
  
• Verification (plug in x values from the table):
  • For x = 3: $y = 10 \cdot (1.5)^{0} = 10$
  • For x = 5: $y = 10 \cdot (1.5)^{2} = 10 \cdot 2.25 = 22.5$
  • The calculator demonstration confirms the values match the table.

Deriving the rule from 22.5 and 75.9375 (two bottom values)

• Consider the two values separated by three steps (from x = 5 to x = 8): 22.5 at x = 5 and 75.9375 at x = 8.
• The relationship is:
  
  75.9375 = 22.5 \cdot r^{3}
  
• Solve for r:
  
  r^{3} = \frac{75.9375}{22.5} = 3.375 \
  r = \sqrt[3]{3.375} = 1.5
  
• Alternative way to compute r: use nth-root function on a calculator (see notes below).
• Anchoring the rule at x = 5 (where y = 22.5):
  
  y = 22.5 \cdot (1.5)^{\,x - 5}
  
• This anchor produces the same table values and demonstrates that you can choose any known point to write the equation.

Using nth-root tools to compute r

• Desmos method:
  • Use the nth root function to compute the root of a ratio, e.g., the cube root of (\dfrac{75.9375}{22.5}).
  • The cube root is written as $\sqrt[3]{\dfrac{75.9375}{22.5}}$ which equals 1.5.
• TI-Nspire method:
  • The cube root (or nth root) button is located above the caret. It shows as an option labeled "nth root x".
  • To compute a 3rd root, enter the index 3 and the radicand, e.g., set up to compute:
    $\sqrt[3]{\dfrac{75.9375}{22.5}}$
  • The result is 1.5, confirming r.
• Practical tip: you can compute r from any pair of known values a and b separated by n steps using:
  
  b = a \cdot r^{n} \quad\Rightarrow\quad r = \sqrt[n]{\dfrac{b}{a}}
  

Equations anchored at different known points (summary)

• If the top value y = 10 occurs at x = 3, then:
  
  y = 10 \cdot (1.5)^{\,x - 3}
  
• If the middle value y = 22.5 occurs at x = 5, then:
  
  y = 22.5 \cdot (1.5)^{\,x - 5}
  
• The same r = 1.5 arises from either pair, reflecting the consistency of the underlying geometric progression.

General concepts and practical implications

• Core idea: exponential growth/decay characterized by a constant ratio r between successive terms.
• The exponent expresses the number of steps away from the chosen anchor x0: $y = y<em>0 \cdot r^{\,x - x</em>0}$ where y0 is the value at x0.
• When x-steps between known values is k, use: $y<em>{new} = y</em>{old} \cdot r^{k}$.
• If the distance between known x-values is d, then find r from: $y<em>{new} = y</em>{old} \cdot r^{d} \Rightarrow r = \sqrt[d]{\dfrac{y<em>{new}}{y</em>{old}}}$.
• Once r is known, you can write the equation from any known point and verify by plugging in other x-values from the table.
• Real-world relevance: modeling exponential growth/decay where the rate of change per step is constant (population growth, compound interest, etc.).
• Additional skills: using calculator tools (Desmos, TI-Nspire) to compute roots (nth roots) and verify equations quickly.

Connections and extensions

• Connects to the concept of a geometric progression: terms have the form yn = y0 r^{n}.
• Connects to the general form of an exponential function: $y = a \cdot r^{x} = a \cdot e^{x \ln r}$ when considering continuous growth.
• If the table uses non-unit x-step sizes, the offset in the exponent accounts for the number of steps between the anchor and the target x-value.

Quick practice prompts (to reinforce)

• Given y = 8 at x = 2 and y = 72 at x = 5, find r and write the equation anchored at x = 2.
• If y = 12 at x = 4 and y = 12 \cdot r^{6} at x = 10, solve for r and express the equation anchored at x = 4.
• Use the nth-root approach to compute r when given y2 = y1 \cdot r^{d} with d = 4 and ratio y2/y1 = 6.25. What is r? (Hint: r = \sqrt[4]{6.25} = 1.5)