Lesson 14 ~25 min Unit 2 · Non-Linear +85 XP

Exponential Relationships $y = a^x$

Doubling, tripling and beyond. Exponentials always pass through $(0, 1)$, hug the $x$-axis on one side, and grow faster than any line or parabola.

Today's hook: $y = 2^x$ at $x = 0$: what is $y$? Why does EVERY exponential $y = a^x$ pass through $(0, 1)$?

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Think First

warm-up

Linear, quadratic, and exponential. Compare $y = 2x$, $y = x^2$, and $y = 2^x$ at $x = 0, 1, 2, 3, 4, 5, 10$. Which grows fastest in the long run? Try to fill in a quick table without a calculator.

Record your answer in your workbook.

The Big Idea

+5 XP

An exponential function $y = a^x$ multiplies by the same factor every time $x$ increases by 1. Constant ratio — not constant difference. The graph always crosses the $y$-axis at $(0, 1)$ because $a^0 = 1$ for any $a > 0$.

The red curve $y = 2^x$ doubles every time $x$ goes up by 1: $(0, 1), (1, 2), (2, 4), (3, 8)$. For $x < 0$, values shrink: $y = 1/2, 1/4, \ldots$ approaching but never touching $y = 0$. The $x$-axis is a horizontal asymptote.

$y = a^x$ — passes $(0, 1)$, horizontal asymptote $y = 0$.

Always $(0, 1)$

$a^0 = 1$ for any $a > 0$.

Constant ratio

Each step $\times a$, not $+ a$.

Asymptote $y = 0$

Curve approaches but never touches the $x$-axis.

What You'll Master

objectives

Know

$y = a^x$ with $a > 1$ is an INCREASING exponential (growth)
Every exponential $y = a^x$ has $y$-intercept $(0, 1)$ and asymptote $y = 0$
Exponentials grow faster than linear or quadratic functions in the long run

Understand

Why $a^0 = 1$ (zero exponent rule) forces the curve through $(0, 1)$
Why the curve never touches the $x$-axis (positive powers can shrink but never reach zero)
Why larger bases give steeper growth (e.g. $3^x$ rises faster than $2^x$)

Can Do

Build a table for $y = 2^x$ or $y = 3^x$
Sketch an exponential and identify its $y$-intercept and asymptote
Compare linear, quadratic and exponential growth rates

Words You Need

vocabulary

Exponential function$y = a^x$, where $a$ (the BASE) is a positive constant.

BaseThe constant being raised to a power — the $a$ in $a^x$.

Exponent (power)The variable $x$ sitting in the "up" position.

Exponential growth$a > 1$: $y$ increases — each step multiplies by $a$.

Horizontal asymptoteThe line $y = 0$ which the curve approaches but never touches.

Zero exponent rule$a^0 = 1$ for any $a > 0$ — forces the curve through $(0, 1)$.

Spot the Trap

heads-up

Wrong: Confusing $y = 2^x$ with $y = x^2$.

Right: $y = x^2$ is a PARABOLA (base is the variable). $y = 2^x$ is an EXPONENTIAL (the variable is the exponent).

Wrong: Saying $y = 2^x$ at $x = 0$ is 0.

Right: $2^0 = 1$, not 0. Every exponential passes through $(0, 1)$.

Building the Table

+5 XP

Table values for $y = 2^x$ at $x = -3, -2, -1, 0, 1, 2, 3$:

$2^{-3} = 1/8$, $2^{-2} = 1/4$, $2^{-1} = 1/2$, $2^0 = 1$, $2^1 = 2$, $2^2 = 4$, $2^3 = 8$.

Pattern: each step right doubles $y$. Each step left halves $y$ — so $y$ shrinks toward zero but never reaches it.

For $y = 3^x$: $1/27, 1/9, 1/3, 1, 3, 9, 27$. Same shape, steeper.

$y = a^x$: each unit step $\times a$ (right) or $\div a$ (left).

Negative $x$ rule

$a^{-n} = 1/a^n$ — gives fractions.

$y$ always positive

$a^x > 0$ for all $x$ — never zero, never negative.

Steeper base, steeper graph

$3^x$ rises faster than $2^x$ for $x > 0$.

Comparing Linear, Quadratic, Exponential

+5 XP

Same $x$, three growth speeds:

Compare $y = 2x$ (linear), $y = x^2$ (quadratic), $y = 2^x$ (exponential):

$x = 0$: $0, 0, 1$.
$x = 2$: $4, 4, 4$ (all equal here!).
$x = 5$: $10, 25, 32$.
$x = 10$: $20, 100, 1024$.
$x = 20$: $40, 400, 1{,}048{,}576$.

For large $x$, the exponential CRUSHES the other two. Linear adds; quadratic squares; exponential multiplies repeatedly — multiplication beats addition in the long run.

Long-term: exponential $\gg$ quadratic $\gg$ linear.

Slow start, runaway finish

Exponentials can look modest, then explode.

Real-world uses

Compound interest, populations, viral spread.

Constant-ratio test

If $y$ multiplies by the same factor each step in $x$ $-$ it's exponential.

Watch Me Solve It · 3 examples

Watch Me Solve It · Table for $y = 2^x$

+15 XP per step

PROBLEM

Complete a table for $y = 2^x$ at $x = -2, -1, 0, 1, 2, 3$. State the $y$-intercept and asymptote.

1
Negative $x$ side
$x = -2: y = 2^{-2} = 1/4$. $x = -1: y = 2^{-1} = 1/2$.
2
Zero and positive $x$
$x = 0: y = 2^0 = 1$. $x = 1: y = 2$. $x = 2: y = 4$. $x = 3: y = 8$.
3
Features
$y$-intercept: $(0, 1)$. Asymptote: $y = 0$.
Each step right doubles; each step left halves. $y$ shrinks toward 0 but never reaches it.

Answer$y$-int $(0, 1)$, asymptote $y = 0$.

Watch Me Solve It · Comparing growth rates

+15 XP per step

PROBLEM

Find $y$ for $y = 2x$, $y = x^2$ and $y = 2^x$ at $x = 4$ and at $x = 10$. State which grows fastest.

1
At $x = 4$
Linear: $2 \times 4 = 8$. Quadratic: $4^2 = 16$. Exponential: $2^4 = 16$. Quadratic and exponential tied.
2
At $x = 10$
Linear: $20$. Quadratic: $100$. Exponential: $2^{10} = 1024$. Exponential pulls ahead.
3
Conclude
Eventually, $y = 2^x$ beats $y = x^2$ and $y = 2x$ for any large enough $x$.
Multiplying repeatedly wins over adding or squaring once $x$ is big enough.

AnswerExponential wins (eventually).

Watch Me Solve It · $2^x$ versus $3^x$

+15 XP per step

PROBLEM

Compare $y = 2^x$ and $y = 3^x$ at $x = 1, 2, 3, 4$. Do they share a $y$-intercept? Which rises faster for $x > 0$?

1
Build tables
$2^x$: $2, 4, 8, 16$. $3^x$: $3, 9, 27, 81$.
2
$y$-intercepts
$2^0 = 1$ and $3^0 = 1$. Both pass $(0, 1)$.
3
Compare growth
At every $x > 0$, $3^x > 2^x$. The larger base grows faster.
For $x < 0$, the larger base shrinks faster instead, so $3^x < 2^x$ when $x < 0$.

AnswerSame $y$-int $(0, 1)$; $3^x$ rises faster for $x > 0$.

Common Pitfalls

heads-up

$a^x$ vs $x^a$

Treating $y = 2^x$ as if it were a parabola, drawing a U-shape.

Fix: variable in the EXPONENT (top) means exponential. Variable in the BASE (bottom) is power/parabola.

$2^0 = 0$ error

Writing $2^0 = 0$, so curve passes through origin.

Fix: $a^0 = 1$ for any $a > 0$. Curve passes through $(0, 1)$, NOT the origin.

Negative $y$ values

Plotting points with negative $y$ for $y = 2^x$.

Fix: $2^x$ is always positive. The curve stays above the $x$-axis.

Copy Into Your Books

Form

$y = a^x$, $a > 0$
Variable is the EXPONENT
$a > 1$: growth

Key features

$y$-int $(0, 1)$ always
Asymptote $y = 0$
$y > 0$ always

Negative $x$

$a^{-n} = 1/a^n$
$y$ approaches 0
Never reaches 0

Growth race

Linear: add $a$
Quadratic: square $x$
Exponential: $\times a$ each step

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Exponential Drills

4 problems

Four quick exponential checks.

1 Find $y$ for $y = 2^x$ at $x = 5$.
$2^5 = 32$.$y = 32$
2 State the $y$-intercept of $y = 3^x$.
$3^0 = 1$.$(0, 1)$
3 Find $y$ for $y = 2^x$ at $x = -3$.
$2^{-3} = 1/2^3$.$y = 1/8$
4 Compare $y = 2x$, $y = x^2$, $y = 2^x$ at $x = 6$.
$12, 36, 64$.Exponential is biggest

Complete in your workbook.

Quick Check · 5 questions

The $y$-intercept of $y = 2^x$ is:

+10 XP

For $y = 2^x$, $y$ at $x = 4$ is:

+10 XP

The horizontal asymptote of $y = 2^x$ is:

+10 XP

For large $x$, which function grows fastest?

+10 XP

For $y = 2^x$, $y$ at $x = -3$ is:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy3 MARKS

Q6. Build a table of values for $y = 3^x$ at $x = -2, -1, 0, 1, 2, 3$. State the $y$-intercept and asymptote.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. Compare $y = 2x$, $y = x^2$ and $y = 2^x$ at $x = 0, 2, 4, 8$. At which value of $x$ does the exponential first exceed both the linear and quadratic, and stay ahead?

Answer in your workbook.

ReasonHard3 MARKS

Q8. A bacterial colony starts with 1 cell and doubles every hour. (a) Write a formula for the number of cells $N$ after $t$ hours. (b) How many cells are present at $t = 5$? (c) Explain why this is an exponential relationship rather than linear.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $2^0 = 1$, so $(0, 1)$.

2. C — $2^4 = 16$.

3. A — asymptote $y = 0$.

4. D — $y = 2^x$ outgrows the others for large $x$.

5. B — $2^{-3} = 1/8$.

Show Your Working Model Answers

Q6 (3 marks): Table: $(-2, 1/9), (-1, 1/3), (0, 1), (1, 3), (2, 9), (3, 27)$ [1]. $y$-intercept $(0, 1)$ [1]. Asymptote $y = 0$ [1].

Q7 (3 marks): $x = 0$: $0, 0, 1$. $x = 2$: $4, 4, 4$. $x = 4$: $8, 16, 16$. $x = 8$: $16, 64, 256$ [1]. At $x = 4$ the exponential equals the quadratic; at $x = 5$ exponential is $32 > 25$ [1]. So the exponential overtakes both around $x = 5$ and stays ahead forever after [1].

Q8 (3 marks): (a) Doubles each hour: $N = 2^t$ [1]. (b) At $t = 5$: $N = 2^5 = 32$ cells [1]. (c) Exponential because each step multiplies by a constant factor (2) rather than adding a constant difference [1].

Stretch Challenge · +25 XP, +10 coins

Exponential Decay — What If $0 < a < 1$?

So far we've used bases $a > 1$ (growth). What about $a < 1$? Consider $y = (1/2)^x$. (a) Build a table for $x = -2, -1, 0, 1, 2, 3$. (b) Does the curve grow or shrink as $x$ increases? (c) Find the $y$-intercept and asymptote. (d) Explain why this is called exponential DECAY and where you'd see it in real life.

Reveal solution

(a) $(1/2)^{-2} = 4$; $(1/2)^{-1} = 2$; $(1/2)^0 = 1$; $(1/2)^1 = 1/2$; $(1/2)^2 = 1/4$; $(1/2)^3 = 1/8$. (b) $y$ shrinks (halves) as $x$ increases by 1 — this is DECAY. (c) $y$-intercept $(0, 1)$ (same as growth); asymptote $y = 0$ (curve approaches but never touches the $x$-axis as $x \to \infty$). (d) Real-world decay: radioactive half-life, drug clearance, asset depreciation, cooling temperatures.

Quick Review

Form

$y = a^x$, $a > 0$

$y$-intercept

$(0, 1)$ always

Asymptote

$y = 0$

$a > 1$

Growth

$0 < a < 1$

Decay

Long-term

Beats lin + quad

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