Increasing exponential
- Input
- 2^x
- Expected output
- f(0) = 1; y = 0
Y = 1 at x = 0; horizontal asymptote y = 0 (as x tends to -∞); increasing. No root: 2^x is never zero.
Carregando ferramenta… / Loading tool…
graph of exponential and logarithmic functions
Type 2^x and then ln(x) or log2(x). The tool draws each curve, dashes the asymptotes, marks the root or the intercept and reports the domain. Plotting the exponential, the logarithm and the line y = x together shows the symmetry between inverses.
Y = 1 at x = 0; horizontal asymptote y = 0 (as x tends to -∞); increasing. No root: 2^x is never zero.
Y = 1 at x = 0; horizontal asymptote y = 0 (as x tends to +∞); decreasing. Mirror image of 2^x across the y axis.
Y = 4 at x = 0; horizontal asymptote y = 3. The constant 3 raises the asymptote.
Root x = 1; vertical asymptote x = 0; defined only for x greater than 0. ln(1) = 0.
Root x = 1; passes through (2, 1) and (8, 3). 2 to the power 3 is 8.
Domain x greater than 2; vertical asymptote x = 2; root x = 3. ln(3 - 2) = ln(1) = 0.
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.
Because a^x is positive for every real x when a is positive: no power of 2 gives zero or a negative number. The curve gets closer and closer to y = 0, for example 2^(-10) = 0.000977, but f(x) = 0 has no solution, so there is no root and the x axis is an asymptote.
log_a(x) is the exponent y such that a^y = x. Since a^y is always positive, x must be positive. That is why ln(x) and log2(x) start at x = 0 and the tool leaves the left side of the graph empty, stating the domain in the panel.
The function reverses direction: (1/2)^x is decreasing and so is log_(1/2)(x). This happens because repeatedly multiplying by a number below 1 shrinks the value. To plot the base 1/2 logarithm type log(x, 0.5) or ln(x)/ln(0.5).
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 |