Graphs of straight lines
A straight line y = mx + c has a gradient m and a y-intercept c. These three lines show a positive gradient, a negative gradient and the line y = x, the mirror line between a function and its inverse.
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 = x
The straight line through the origin with gradient 1: every point has equal x- and y-coordinates. It is the line of symmetry between a function and its inverse.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | all real numbers (y ∈ ℝ) |
| x-intercept | (0, 0) |
| y-intercept | (0, 0) |
| Turning points | none |
| Asymptotes | none |
| Gradient | 1 |
| Symmetry | odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin |
Reflecting a graph in y = x gives the graph of the inverse function.
How to sketch it
- Mark the x-intercept (0, 0), the y-intercept (0, 0).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
- Use the symmetry: draw the right-hand half and rotate it 180° about the origin.
Graphs made from this one: y = 2x + 1, y = 3 − 2x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = 2x + 1
A straight line with gradient 2 and y-intercept 1: y goes up 2 for every 1 across.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | all real numbers (y ∈ ℝ) |
| x-intercept | (−1/2, 0) |
| y-intercept | (0, 1) |
| Turning points | none |
| Asymptotes | none |
| Gradient | 2 |
| Symmetry | neither even nor odd |
How to sketch it
- Mark the x-intercept (−1/2, 0), the y-intercept (0, 1).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
y = 2x + 1 is y = x stretched vertically by scale factor 2, then translated up 1: see the graph of y = x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Graph of y = 3 − 2x
A straight line with a negative gradient, −2: y goes down 2 for every 1 across, and it crosses the y-axis at 3.
Key features
| Domain | all real numbers (x ∈ ℝ) |
|---|---|
| Range | all real numbers (y ∈ ℝ) |
| x-intercept | (3/2, 0) |
| y-intercept | (0, 3) |
| Turning points | none |
| Asymptotes | none |
| Gradient | −2 |
| Symmetry | neither even nor odd |
How to sketch it
- Mark the x-intercept (3/2, 0), the y-intercept (0, 3).
- The domain is every real number, so the curve continues off both sides of your axes.
- Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
y = 3 − 2x is y = x reflected in the y-axis, stretched vertically by scale factor 2, then translated up 3: see the graph of y = x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.
Notes and more graphs
Other families of graphs
- 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 exponential and logistic 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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