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 valueswhere 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 anchorit 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 confidencethat 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.
Mark lesson as complete
Tick when you've finished the practice and review.