Mathematics • Year 9 • Unit 2 • Lesson 14
Exponentials — Mixed Challenge
Pull together every exponential skill: read values (including negative exponents), compare bases, contrast with linear and quadratic, and meet exponential DECAY for $0 < a < 1$. Then catch a confused-base mistake and design your own growth story.
1. Mixed problems
Show working. 3 marks each
1.1 Find $y$ for each of: (a) $y = 2^x$ at $x = 6$ (b) $y = 3^x$ at $x = 4$ (c) $y = 5^x$ at $x = 0$ (d) $y = 2^x$ at $x = -4$.
1.2 Build a table for $y = 5^x$ at $x = -1, 0, 1, 2, 3$. State the $y$-intercept and asymptote. Then state whether the curve is steeper or flatter than $y = 2^x$ for $x > 0$.
1.3 Compare $y = 2^x$ and $y = 3^x$ at $x = -2, 0, 2, 4$. (a) State the value of each at $x = 0$. (b) Which is bigger for $x > 0$? (c) Which is bigger for $x < 0$?
1.4 Sort each into the correct family: (a) $y = 2x$ (b) $y = x^2$ (c) $y = 2^x$ (d) $y = 5x + 1$ (e) $y = (\tfrac{1}{2})^x$. For each, name the family and state the value of $y$ at $x = 3$.
1.5 Exponential decay. For $y = (\tfrac{1}{2})^x$: (a) build a table at $x = -2, -1, 0, 1, 2, 3$, (b) state the $y$-intercept and asymptote, (c) explain why this is called "decay" (not "growth").
1.6 A culture starts with $5$ cells and doubles every hour. (a) Write a formula for the cell count $N$ at hour $t$. (b) When does the count first exceed $200$? (c) Why is this "exponential" rather than linear?
2. Find the mistake
A student is asked to evaluate four expressions. Two of their answers are wrong. Spot, explain, and fix. 3 marks
Student's working:
A: $2^5 = 10$ (because $2 \times 5 = 10$)
B: $3^0 = 1$ ✓
C: $2^{-3} = -8$ (because $-3$ is negative)
D: $5^2 = 25$ ✓
(a) Which two are wrong?
(b) For each wrong one, explain in ONE sentence the rule the student broke.
(c) Give the correct values for the two wrong ones.
Stuck? $2^5$ means $2 \times 2 \times 2 \times 2 \times 2 = 32$ (repeated multiplication, not multiplication of $2$ by $5$). $2^{-3} = 1/2^3 = 1/8$ (negative exponent flips to reciprocal, not to a negative number).3. Open-ended challenge — design a growth story
This question has many valid answers. Be creative. 4 marks
3.1 Invent a real-world scenario (DIFFERENT from rumours, bacteria, viral views, tournaments and savings) where a quantity $y$ grows exponentially over time with base $a$, starting value $y_0$, and formula $y = y_0 \cdot a^t$.
For your story:
(i) Describe the situation in 1–2 sentences and clearly name what $y$ is (units) and what $t$ is.
(ii) State your starting value $y_0$ and growth multiplier $a$ (you choose any $a > 1$).
(iii) Write the formula.
(iv) Build a table for $t = 0, 1, 2, 3, 4, 5$.
(v) Predict what value $y$ first crosses: e.g. "first crosses 1000" or whatever round number suits your story.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Single values
(a) $2^6 = 64$. (b) $3^4 = 81$. (c) $5^0 = 1$. (d) $2^{-4} = 1/2^4 = 1/16$.
1.2 — Table for $y = 5^x$
$x = -1, 0, 1, 2, 3$ gives $y = 1/5, 1, 5, 25, 125$. $y$-intercept $(0, 1)$. Asymptote $y = 0$. For $x > 0$, $5^x$ rises MUCH steeper than $2^x$ — bigger base, faster growth.
1.3 — $2^x$ vs $3^x$
(a) Both equal $1$ at $x = 0$.
(b) For $x > 0$, $3^x > 2^x$ (e.g. at $x = 2$: $9 > 4$). Bigger base wins on the positive side.
(c) For $x < 0$, $3^x < 2^x$ (e.g. at $x = -2$: $1/9 < 1/4$). Bigger base shrinks faster on the negative side.
1.4 — Sort into families
(a) Linear, $y(3) = 6$. (b) Quadratic (parabola), $y(3) = 9$. (c) Exponential, $y(3) = 8$. (d) Linear, $y(3) = 16$. (e) Exponential (decay), $y(3) = (1/2)^3 = 1/8$.
1.5 — Exponential decay $y = (1/2)^x$
(a) $x = -2, -1, 0, 1, 2, 3$ gives $y = 4, 2, 1, 1/2, 1/4, 1/8$.
(b) $y$-intercept $(0, 1)$. Asymptote $y = 0$.
(c) Called "decay" because as $x$ increases, $y$ DECREASES (not increases). The base $\tfrac{1}{2}$ is less than $1$, so each step right multiplies by $\tfrac{1}{2}$ — halving each time.
1.6 — Doubling colony
(a) $\mathbf{N = 5 \times 2^t}$.
(b) $N(5) = 5 \times 32 = 160$ (still $\le 200$). $N(6) = 5 \times 64 = 320$ (over). So $N$ first exceeds $200$ at $t = 6$ hours.
(c) Exponential because the count is MULTIPLIED by the same factor ($2$) each hour, not added to by a fixed amount. Linear would be "+5 cells per hour"; exponential is "$\times 2$ per hour".
2 — Find the mistake
(a) Wrong ones are A and C.
(b) A: the student treated $2^5$ as $2 \times 5$, but the exponent means REPEATED MULTIPLICATION ($2 \times 2 \times 2 \times 2 \times 2$), not multiplying $2$ by the exponent. C: the student let the negative exponent make the answer negative, but a NEGATIVE EXPONENT flips the base to its reciprocal — it does NOT make the result negative.
(c) A: $2^5 = 32$. C: $2^{-3} = 1/2^3 = 1/8$.
3 — Open-ended challenge (sample solution)
Story: Lily pads on a pond. A pond starts with $2$ lily pads, and the number triples each week.
(i) $y$ = number of lily pads (count), $t$ = weeks since the start.
(ii) $y_0 = 2$, $a = 3$.
(iii) Formula: $\mathbf{y = 2 \times 3^t}$.
(iv) Table: $t = 0, 1, 2, 3, 4, 5$ gives $y = 2, 6, 18, 54, 162, 486$.
(v) Crosses 100 between $t = 3$ ($54$) and $t = 4$ ($162$). Crosses 500 between $t = 4$ ($162$) and $t = 5$ ($486$ — just under) — would cross at the start of $t = 5$ roughly.
Marking: 1 mark for a distinct story; 1 mark for valid $y_0$ and $a > 1$; 1 mark for correct table; 1 mark for a sensible threshold prediction.