Increasing line
- Input
- 2x + 1
- Expected output
- x = -0.5; f(0) = 1; a = 2
Root -0.5; y = 1 at x = 0; increasing. a = 2, b = 1.
Carregando ferramenta… / Loading tool…
graph a linear function y = mx + b
Write the function, for example 2x + 1, and the line appears with its root and the point where it crosses the y axis marked. To compare lines, add more functions: the panel tells you where they cross or whether they are parallel.
Root -0.5; y = 1 at x = 0; increasing. a = 2, b = 1.
Root 4; y = 4 at x = 0; decreasing. a = -1.
Root 6; y = -3 at x = 0. -b/a = 3/0.5 = 6.
Root 0; passes through the origin. b = 0, direct variation.
No root; horizontal line y = 5. a = 0, the graph neither rises nor falls.
Intersection (0.8, 2.6). 2 times -1/2 is -1.
Type the function, for example x^2 - 4x + 3. The parabola opens upward if a is positive and downward if it is negative. The vertex is at x = -b/2a: here x = 2 and y = -1. The roots come from (-b ± √Δ)/2a with Δ = b² - 4ac = 4, so x = 1 and x = 3, and the parabola crosses the y axis at c = 3. The tool marks all of these points and shows their exact form.
Compute a = (y2 - y1)/(x2 - x1) and then b = y1 - a·x1. For (1, 2) and (3, 8): a = 6/2 = 3 and b = 2 - 3 = -1, so y = 3x - 1. Type the equation in the tool and confirm that the line passes through the points.
It is the number a in y = ax + b and measures how steep the line is. With a = 1 the line makes a 45 degree angle with the x axis; with a = 3 it rises three times as fast; with a = -1 it falls at the same rate. Use the sliders by typing a*x + b to watch the line rotate around the point (0, b).
When they have the same slope and different intercepts, that is, they are distinct parallel lines: 2x + 1 and 2x + 4, for example. If they also have the same b they are the same line and every point coincides, a case the tool also recognizes.
Quadratic function, f(x) = ax² + bx + c with a not equal to 0. The graph is a parabola, symmetric about the vertical line through the vertex. Here a = 1, b = -4 and c = 3; the parabola opens upward.
Exact result (algebra)
Linear function, f(x) = ax + b. The graph is a line: a is the slope (increasing if a is positive) and b is where the line crosses the y axis. Here a = 0.5 (increasing) and b = 1.
Exact result (algebra)
| x | f₁(x) | f₂(x) |
|---|---|---|
| -5 | 48 | -1.5 |
| -4 | 35 | -1 |
| -3 | 24 | -0.5 |
| -2 | 15 | 0 |
| -1 | 8 | 0.5 |
| 0 | 3 | 1 |
| 1 | 0 | 1.5 |
| 2 | -1 | 2 |
| 3 | 0 | 2.5 |
| 4 | 3 | 3 |
| 5 | 8 | 3.5 |