Graphs of Exponential Functions
The shape of an exponential graph reveals its behaviour at a glance: always positive, always passing through $(0,1)$, with a horizontal asymptote the curve approaches but never reaches. Transformations shift, stretch, and flip this shape — but the structural anchors remain.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
Sketch your gut version of $y = 2^x$ on paper. Without using a table of values — where does it cross the $y$-axis, and where does the curve flatten out toward as $x$ becomes very negative?
Every exponential sketching question follows the same two-step attack: identify the key features first, then draw the curve through them. Lock the feature-reading rules into muscle memory and the sketch almost draws itself.
For $y = a^{x-h} + k$: the asymptote moves to $y = k$, and the reference point (where the basic curve has value 1) moves to $(h,\ 1+k)$. Everything else follows from these two anchors.
Key facts
- Key features of $y = a^x$: intercept, asymptote, domain, range
- How each transformation parameter affects the graph
- That $y = a^{-x} = \left(\frac{1}{a}\right)^x$
Concepts
- Why the asymptote never equals $y = 0$ when $k \neq 0$
- How the $y$-axis reflection turns growth into decay
- The structural anchors of any exponential graph
Skills
- Sketch any transformed exponential with labelled features
- State the range from the asymptote and any reflections
- Find the equation of an exponential from two given points
The basic exponential graph $y = a^x$ has these features: it always lies above the $x$-axis (range $y > 0$), it passes through $(0, 1)$, and the $x$-axis ($y = 0$) is a horizontal asymptote. For $a > 1$, the graph rises steeply to the right. For $0 < a < 1$, the graph falls to the right.
Transformations follow the same rules as for other functions. The asymptote is the structural anchor — it moves with any vertical translation but stays horizontal. When labelling a sketch, always draw the asymptote first as a dashed line, then plot the $y$-intercept, then draw the curve through these features.
Vertical translation moves the asymptote. Reflection in $x$-axis flips the curve below the asymptote.
Basic $y = a^x$: asymptote $y = 0$; $y$-intercept $(0,1)$; range $y > 0$; $y = a^{x-h} + k$: asymptote $y = k$; passes through $(h, 1+k)$; range $y > k$
Pause — copy the two feature sets: basic $y = a^x$ (asymptote $y = 0$, passes through $(0,1)$, range $y > 0$) and shifted $y = a^{x-h} + k$ (asymptote $y = k$, passes through $(h, 1+k)$) into your book.
Did you get this? True or false: the horizontal asymptote of $y = 3^x - 5$ is $y = 0$.
Worked examples · 3 in a row, reveal as you go
Sketch $y = 2^x$ showing all key features.
Sketch $y = 3^{x-1} + 2$ and state its range.
Find the equation of an exponential curve passing through $(0, 3)$ and $(1, 6)$.
Quick check: What is the horizontal asymptote of $y = 2^{x+3} - 4$?
Common errors · the 3 traps that cost marks
Odd one out — Three of these are key features you would label on a sketch of $y = a^x$. Which one does NOT belong?
Quick-fire practice · 5 problems
Sketch $y = \left(\frac{1}{2}\right)^x$ showing the asymptote and $y$-intercept.
State the horizontal asymptote and range of $y = 2^x - 3$.
Describe the transformation from $y = 3^x$ to $y = 3^{x+2}$. Is it left or right?
Sketch $y = -2^x$ and state its range and asymptote.
Find the equation of $y = a^x$ passing through $(0, 1)$ and $(2, 9)$.
Did you get this? True or false: the range of $y = -3^x + 5$ is $y < 5$.
Match up — Drag or identify which description matches each equation.
Earlier you sketched $y = 2^x$ from memory. The curve crosses at $(0, 1)$, rises steeply to the right, and flattens toward the horizontal asymptote $y = 0$ as $x \to -\infty$. Transformations of the form $y = A \cdot b^{x-h} + k$ shift, stretch, or flip this basic shape — but the asymptote and $y$-intercept remain the structural anchors that always appear on your sketch.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Sketch $y = 2^{x-1} + 1$, labelling the asymptote, $y$-intercept, and one other point. (3 marks)
Q2. An exponential curve has equation $y = a \cdot b^x$ and passes through $(0, 5)$ and $(2, 20)$. Find $a$ and $b$. (3 marks)
Q3. Describe the sequence of transformations that maps $y = 2^x$ onto $y = -2^{x+1} + 3$. State the range of the transformed function. (4 marks)
Comprehensive answers (click to reveal)
Drill answers: 1) decay curve; asymptote $y=0$; intercept $(0,1)$ · 2) asymptote $y = -3$; range $y > -3$ · 3) shift left 2 units · 4) range $y < 0$; asymptote $y = 0$ · 5) $y = 3^x$ (since $a^2 = 9 \Rightarrow a = 3$)
Q1 (3 marks): Asymptote: $y = 1$ [0.5]. $y$-intercept at $x = 0$: $y = 2^{-1} + 1 = \frac{1}{2} + 1 = \frac{3}{2}$, so $(0, \frac{3}{2})$ [1]. At $x = 1$: $y = 2^0 + 1 = 2$, so $(1, 2)$ [0.5]. Correct shape: increasing curve approaching $y = 1$ from above [1].
Q2 (3 marks): At $(0, 5)$: $5 = a \cdot b^0 = a$, so $a = 5$ [1]. At $(2, 20)$: $20 = 5 \cdot b^2$, so $b^2 = 4$ [1]. $b = 2$ (since $b > 0$) [1].
Q3 (4 marks): $x+1$ inside the exponent: shift left 1 unit [1]. Negative sign outside: reflect in the $x$-axis [1]. $+3$ outside: shift up 3 units [1]. Original range $y > 0$; after reflection $y < 0$; after shift up 3: $y < 3$ [1].
Five timed questions. Beat the boss to bank a tier — gold (90%+ speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering exponential graph questions. A lighter alternative to the boss.
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