Mathematics • Year 9 • Unit 2 • Lesson 14
Exponential Relationships $y = a^x$
Build the table-then-features habit for exponentials. Watch one worked example for $y = 2^x$, fill in a guided one for $y = 3^x$, then run eight independent problems — including negative exponents.
1. I do — fully worked example
Read every line. Watch how the table is built on BOTH sides of $x = 0$ (positive and negative exponents).
Problem. Build a table of values for $y = 2^x$ at $x = -2, -1, 0, 1, 2, 3$. State the $y$-intercept and the horizontal asymptote.
Step 1 — Negative exponents.
$x = -2$: $y = 2^{-2} = 1/2^2 = 1/4$. $x = -1$: $y = 2^{-1} = 1/2$.
Reason: a negative exponent flips the base to its reciprocal. $2^{-n} = 1/2^n$.
Step 2 — Zero exponent.
$x = 0$: $y = 2^0 = 1$.
Reason: $a^0 = 1$ for ANY positive base. This forces the curve through $(0, 1)$.
Step 3 — Positive exponents.
$x = 1$: $y = 2$. $x = 2$: $y = 4$. $x = 3$: $y = 8$.
Reason: each step right multiplies by the base. Doubling, doubling, doubling.
Step 4 — Read off the features.
$y$-intercept = $(0, 1)$. As $x \to -\infty$, $y \to 0$ but never reaches it: horizontal asymptote $y = 0$.
Answer: Table: $(-2, \tfrac{1}{4}), (-1, \tfrac{1}{2}), (0, 1), (1, 2), (2, 4), (3, 8)$. $y$-int $(0, 1)$, asymptote $y = 0$.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. Build a table for $y = 3^x$ at $x = -2, -1, 0, 1, 2, 3$. State the $y$-intercept and the asymptote.
Step 1 — Negative exponents: $3^{-2} = 1/3^{\_\_} = $ ______ (a fraction). $3^{-1} = $ ______ (a fraction).
Step 2 — Zero exponent: $3^0 = $ ______ . The curve passes through $($ ______ $,$ ______ $)$.
Step 3 — Positive exponents: $3^1 = $ ______ . $3^2 = $ ______ . $3^3 = $ ______ .
Step 4 — Features: $y$-intercept is $($ ______ $,$ ______ $)$. Horizontal asymptote: $y = $ ______ (which line does the curve hug as $x \to -\infty$?).
3. You do — independent practice
Show your working under each problem. 3.1–3.4 are foundation (single values). 3.5–3.6 are standard (mini-tables and features). 3.7–3.8 are extension (comparisons and modelling).
Foundation — single values
3.1 Find $y$ for $y = 2^x$ at $x = 5$. 1 mark
3.2 State the $y$-intercept of $y = 5^x$. 1 mark
3.3 Find $y$ for $y = 2^x$ at $x = -3$. 1 mark
3.4 State the horizontal asymptote of $y = 7^x$. 1 mark
Standard — tables and features
3.5 Build a table of values for $y = 3^x$ at $x = -2, -1, 0, 1, 2, 3$. State the $y$-intercept and the horizontal asymptote. 2 marks
3.6 Build a table for $y = 4^x$ at $x = -1, 0, 1, 2$. State the $y$-intercept. Then explain in one sentence why $y$ is never negative for $y = 4^x$. 2 marks
Extension — comparisons and modelling
3.7 Compare $y = 2x$, $y = x^2$ and $y = 2^x$ at $x = 0, 2, 4, 6, 8$. At which value of $x$ does the exponential first overtake BOTH the linear and quadratic, and stay ahead? 2 marks
3.8 A bacterial colony starts with $1$ cell and doubles every hour. Let $t$ = hours and $N$ = number of cells. (a) Write a formula for $N$ in terms of $t$. (b) How many cells are there at $t = 5$ hours? (c) Explain in one sentence why this is exponential, not linear. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $y = 3^x$)
Step 1: $3^{-2} = 1/3^{\mathbf{2}} = \mathbf{1/9}$. $3^{-1} = \mathbf{1/3}$.
Step 2: $3^0 = \mathbf{1}$. Curve through $(\mathbf{0}, \mathbf{1})$.
Step 3: $3^1 = \mathbf{3}$. $3^2 = \mathbf{9}$. $3^3 = \mathbf{27}$.
Step 4: $y$-intercept $(\mathbf{0}, \mathbf{1})$. Asymptote $y = \mathbf{0}$ (the $x$-axis).
3.1 — $y = 2^x$ at $x = 5$
$2^5 = 32$. So $y = \mathbf{32}$.
3.2 — $y$-intercept of $y = 5^x$
$5^0 = 1$, so $y$-intercept $= \mathbf{(0, 1)}$. (True of every $y = a^x$ with $a > 0$.)
3.3 — $y = 2^x$ at $x = -3$
$2^{-3} = 1/2^3 = \mathbf{1/8}$.
3.4 — Asymptote of $y = 7^x$
$\mathbf{y = 0}$ (the $x$-axis). True of every $y = a^x$ with $a > 1$.
3.5 — Table for $y = 3^x$
$x = -2, -1, 0, 1, 2, 3$ gives $y = 1/9, 1/3, 1, 3, 9, 27$.
$y$-intercept $(0, 1)$. Asymptote $y = 0$.
3.6 — Table for $y = 4^x$
$x = -1, 0, 1, 2$ gives $y = 1/4, 1, 4, 16$.
$y$-intercept $(0, 1)$. $y$ is never negative because $4^x$ is a positive base raised to any power — the result is always positive.
3.7 — Compare $2x$, $x^2$, $2^x$
Table at $x = 0, 2, 4, 6, 8$:
$2x$: $0, 4, 8, 12, 16$.
$x^2$: $0, 4, 16, 36, 64$.
$2^x$: $1, 4, 16, 64, 256$.
At $x = 0$ and $x = 2$: ties (or exponential equal/larger). At $x = 4$: $2^x = 16$ ties $x^2 = 16$, both ahead of $2x = 8$. At $x = 6$: $2^x = 64$ is the biggest. From $x = 6$ onwards the exponential is ahead and stays ahead. So the exponential first overtakes both at $x = 6$.
3.8 — Bacterial colony
(a) $\mathbf{N = 2^t}$ (since 1 cell doubling each hour: $1 \to 2 \to 4 \to 8 \to \ldots$).
(b) At $t = 5$: $N = 2^5 = \mathbf{32}$ cells.
(c) Exponential, not linear, because each hour MULTIPLIES the count by 2 rather than ADDING a fixed number — the growth rate gets bigger as the colony gets bigger.