Graphs of exponential and logistic functions
A graph y = aˣ grows or decays by the same factor over equal steps in x and approaches the x-axis on one side. The logistic curve grows exponentially at first, then levels off at a limit.
Every value in the key-features tables is checked by script against the Grapher's own calculations (roots, turning points, asymptotes, domain and range).
Graph of y = eˣ
The exponential function: always positive, crossing the y-axis at 1, growing faster and faster to the right and approaching the x-axis to the left. Its gradient at every point equals its height.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | y > 0 |
| x-intercepts | none |
| y-intercept | (0, 1) |
| Turning points | none |
| Asymptotes | horizontal: y = 0 (the x-axis), as x → −∞ |
| Symmetry | neither even nor odd |
How to sketch it
- Draw the asymptote first as dashed lines: y = 0 (the x-axis), as x → −∞.
- Mark the y-intercept (0, 1).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: y > 0. Your sketch should not go anywhere else.
Graphs made from this one: y = e⁻ˣ, y = 2ˣ, y = ln x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = e⁻ˣ
Exponential decay: the reflection of y = eˣ in the y-axis. It falls towards the x-axis as x increases.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | y > 0 |
| x-intercepts | none |
| y-intercept | (0, 1) |
| Turning points | none |
| Asymptotes | horizontal: y = 0 (the x-axis), as x → ∞ |
| Symmetry | neither even nor odd |
How to sketch it
- Draw the asymptote first as dashed lines: y = 0 (the x-axis), as x → ∞.
- Mark the y-intercept (0, 1).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: y > 0. Your sketch should not go anywhere else.
y = e⁻ˣ is y = eˣ reflected in the y-axis: see the graph of y = eˣ. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = 2ˣ
Exponential growth with base 2: the value doubles every time x goes up by 1.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | y > 0 |
| x-intercepts | none |
| y-intercept | (0, 1) |
| Turning points | none |
| Asymptotes | horizontal: y = 0 (the x-axis), as x → −∞ |
| Symmetry | neither even nor odd |
How to sketch it
- Draw the asymptote first as dashed lines: y = 0 (the x-axis), as x → −∞.
- Mark the y-intercept (0, 1).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: y > 0. Your sketch should not go anywhere else.
y = 2ˣ is 2ˣ = e^(x ln 2): y = eˣ stretched horizontally by scale factor 1/ln 2 ≈ 1.44: see the graph of y = eˣ. Graphs made from this one: y = (1/2)ˣ, y = log₂ x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = (1/2)ˣ
Exponential decay with base 1/2: the value halves every time x goes up by 1. It is the same graph as y = 2⁻ˣ.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | y > 0 |
| x-intercepts | none |
| y-intercept | (0, 1) |
| Turning points | none |
| Asymptotes | horizontal: y = 0 (the x-axis), as x → ∞ |
| Symmetry | neither even nor odd |
How to sketch it
- Draw the asymptote first as dashed lines: y = 0 (the x-axis), as x → ∞.
- Mark the y-intercept (0, 1).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: y > 0. Your sketch should not go anywhere else.
y = (1/2)ˣ is y = 2ˣ reflected in the y-axis: see the graph of y = 2ˣ. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = e^(−x²)
A bell-shaped curve (the shape of the normal distribution) with its maximum at (0, 1).
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | 0 < y ≤ 1 |
| x-intercepts | none |
| y-intercept | (0, 1) |
| Turning point | maximum (0, 1) |
| Points of inflection | (−√2/2, e^(−1/2)) ≈ (−0.707, 0.607); (√2/2, e^(−1/2)) ≈ (0.707, 0.607) |
| Asymptotes | horizontal: y = 0 (the x-axis) |
| Symmetry | even function: f(−x) = f(x), symmetric about the y-axis |
How to sketch it
- Draw the asymptote first as dashed lines: y = 0 (the x-axis).
- Mark the y-intercept (0, 1), the turning point (0, 1).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: 0 < y ≤ 1. Your sketch should not go anywhere else.
- Use the symmetry: draw the right-hand half and reflect it in the y-axis.
Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = xeˣ
A product of x and eˣ. The product rule gives dy/dx = (x + 1)eˣ, so the minimum is at x = −1.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | y ≥ −1/e (≈ −0.368) |
| x-intercept | (0, 0) |
| y-intercept | (0, 0) |
| Turning point | minimum (−1, −1/e) ≈ (−1, −0.368) |
| Point of inflection | (−2, −2/e²) ≈ (−2, −0.271) |
| Asymptotes | horizontal: y = 0 (the x-axis), as x → −∞ |
| Symmetry | neither even nor odd |
How to sketch it
- Draw the asymptote first as dashed lines: y = 0 (the x-axis), as x → −∞.
- Mark the x-intercept (0, 0), the y-intercept (0, 0), the turning point (−1, −1/e) ≈ (−1, −0.368).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: y ≥ −1/e (≈ −0.368). Your sketch should not go anywhere else.
Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = 1/(1 + e⁻ˣ)
The logistic (sigmoid) curve: an S shape between the asymptotes y = 0 and y = 1, growing fastest at its point of inflection (0, 1/2).
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | 0 < y < 1 |
| x-intercepts | none |
| y-intercept | (0, 1/2) |
| Turning points | none |
| Point of inflection | (0, 1/2) |
| Asymptotes | horizontal: y = 0, as x → −∞; horizontal: y = 1, as x → ∞ |
| Symmetry | neither even nor odd |
It has rotational symmetry of order 2 about (0, 1/2).
How to sketch it
- Draw the asymptotes first as dashed lines: y = 0, as x → −∞; y = 1, as x → ∞.
- Mark the y-intercept (0, 1/2).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: 0 < y < 1. Your sketch should not go anywhere else.
Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Notes and more graphs
- Exponential and log functions (AA SL notes)
- Exponential models (AI SL notes)
- Modelling with functions (AI HL notes)
- Logistic model for your IA
Other families of graphs
- Graphs of straight lines
- Graphs of quadratic functions
- Graphs of cubic and quartic functions
- Graphs of square root and cube root functions
- Graphs of reciprocal and rational functions
- Graphs of modulus (absolute value) functions
- Graphs of logarithmic functions
- Graphs of trigonometric functions
- Graphs of inverse trigonometric functions
All graphs of common functions · Graph transformations · Open the Grapher
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