Online Function Grapher

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graph of sine and cosine

Graph of sine and cosine: amplitude, period and shift

Type sin(x) and cos(x) and turn on the x axis in multiples of π. To study the wave, type a*sin(b*x) + c and move the sliders: a changes the height, b changes the frequency and c raises or lowers the whole wave.

Amplitude, period and roots of the wave

The sine and cosine functions are defined for every real x, always take values between -1 and 1 and repeat every 2π, about 6.283. Sine passes through the origin (sin 0 = 0) and is an odd function, symmetric about the origin. Cosine equals 1 at x = 0 and is an even function, symmetric about the y axis. The cosine graph is the sine graph shifted π/2 to the left, since cos(x) = sin(x + π/2). The tool computes angles in radians, the standard unit in calculus.
In f(x) = a·sin(bx) + c, the coefficient a sets the amplitude |a|: 2sin(x) oscillates between -2 and 2. The coefficient b sets the period 2π/|b|: sin(2x) has period π and completes two waves in the span of one wave of sin(x), while sin(x/2) has period 4π. The term c shifts the wave vertically and the range of values becomes c - |a| to c + |a|: sin(x) + 1 runs from 0 to 2 and is never negative. Typing a*sin(b*x) + c lets you see each effect separately.
The roots of sine are the multiples of π: ..., -π, 0, π, 2π, ... Between -10 and 10 there are seven (from -3π to 3π). The roots of cosine sit at π/2 plus multiples of π. The maxima of sine (value 1) occur at π/2 + 2kπ and the minima (value -1) at 3π/2 + 2kπ, and the tool marks them all as diamonds. The tangent, tan(x) = sin(x)/cos(x), has vertical asymptotes where cosine is zero, at π/2 + kπ, and period π instead of 2π.
Turn on x axis in multiples of π to read the labels π/2, π, 3π/2 and 2π instead of decimals like 1.571 and 3.142. Since waves are periodic, the properties panel estimates roots and extrema only over the region shown, and flags the result as an estimate. If you zoom out and see more periods, the list grows with it. Sine and cosine cross at π/4 + kπ, where both equal ±√2/2, about ±0.7071, and the intersections list confirms it.

Checked examples

Sine

Input: sin(x)
Expected output: f(0) = 0; f(π/2) = 1; f(π) = 0

Roots at multiples of π (seven between -10 and 10); maximum at (π/2, 1); odd function. Period 2π, amplitude 1.

Cosine

Input: cos(x)
Expected output: f(0) = 1; f(π/2) = 0

Y = 1 at x = 0; even function; first positive root at π/2. cos(x) = sin(x + π/2).

Amplitude 2

Input: 2*sin(x)
Expected output: [-2, 2]

Maximum 2 and minimum -2; same roots as sine. Amplitude |a| = 2.

Doubled frequency

Input: sin(2*x)
Expected output: T = π; f(π/2) = 0

Period π; roots at multiples of π/2. Period 2π/2 = π.

Shifted wave

Input: sin(x) + 1
Expected output: [0, 2]; f(0) = 1

Values from 0 to 2; y = 1 at x = 0. c = 1 raises the wave.

Sine and cosine together

Input: sin(x) e cos(x)
Expected output: (π/4, 0.7071)

First positive intersection at (π/4, 0.7071). Both equal √2/2.

Full tool FAQ

How do I graph a quadratic function?

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.

How do I find the roots of a function on the graph?

How do I identify the asymptotes of a function?

What is the difference between an asymptote and a hole in a graph?

How do I plot several graphs and find where they cross?

What are the parameters a, b and c?

Which functions and symbols can I use?

Does the panel show the important points even off screen?

How do I save or share the graph?

Why does my function not show up or give an error?

Frequently asked questions

How do I compute the period of a sine function?

The period of a·sin(bx + d) + c is 2π/|b|. In sin(3x) it is 2π/3, about 2.094: the wave repeats three times in the space sin(x) uses for one. Only b changes the period; a, c and d change amplitude, height and horizontal shift.

Why does the graph use radians and not degrees?

Because calculus and most advanced math use radians: sin(π) = 0, but sin(180) in radians is about -0.801. To plot in degrees, convert: sin(x·π/180) draws the sine with x in degrees, and then the period is 360 instead of 2π.

Do sine and cosine have asymptotes?

No. Both are continuous and bounded between -1 and 1, with no vertical asymptote, and they oscillate instead of approaching a line, so there is no horizontal asymptote. The tangent is the one with vertical asymptotes, at the points where cosine is zero, and the tool marks all of them.

Next steps

Properties

f₁(x) = x^2 - 4x + 3Quadratic

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)

Domain: ℝ
Range: [-1, +∞)
Roots (zeros): x = 1
x = 3
y intercept: (0, 3)
Vertex: (2, -1) minimum point
Axis of symmetry: x = 2
Discriminant Δ: 4 (two real roots)
Increasing and decreasing: Decreasing on (-∞, 2]
Increasing on [2, +∞)
Sign: f₁(x) > 0 on (-∞, 1) ∪ (3, +∞)
f₁(x) < 0 on (1, 3)
Asymptotes: None found
Parity: Neither even nor odd.

f₂(x) = x/2 + 1Linear (1st degree)

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)

Domain: ℝ
Range: ℝ
Roots (zeros): x = -2
y intercept: (0, 1)
Increasing and decreasing: Increasing on (-∞, +∞)
Sign: f₂(x) > 0 on (-2, +∞)
f₂(x) < 0 on (-∞, -2)
Asymptotes: None found
Parity: Neither even nor odd.

Intersections between the functions

f₁ ∩ f₂ (0.5, 1.25)
f₁ ∩ f₂ (4, 3)

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

f₁(x)

f₂(x)

-5

-1.5

-4

-1

-3

-0.5

-2

-1

0.5

1.5

-1

2.5

3.5

Amplitude, period and roots of the wave

The sine and cosine functions are defined for every real x, always take values between -1 and 1 and repeat every 2π, about 6.283. Sine passes through the origin (sin 0 = 0) and is an odd function, symmetric about the origin. Cosine equals 1 at x = 0 and is an even function, symmetric about the y axis. The cosine graph is the sine graph shifted π/2 to the left, since cos(x) = sin(x + π/2). The tool computes angles in radians, the standard unit in calculus.

In f(x) = a·sin(bx) + c, the coefficient a sets the amplitude |a|: 2sin(x) oscillates between -2 and 2. The coefficient b sets the period 2π/|b|: sin(2x) has period π and completes two waves in the span of one wave of sin(x), while sin(x/2) has period 4π. The term c shifts the wave vertically and the range of values becomes c - |a| to c + |a|: sin(x) + 1 runs from 0 to 2 and is never negative. Typing a*sin(b*x) + c lets you see each effect separately.

The roots of sine are the multiples of π: ..., -π, 0, π, 2π, ... Between -10 and 10 there are seven (from -3π to 3π). The roots of cosine sit at π/2 plus multiples of π. The maxima of sine (value 1) occur at π/2 + 2kπ and the minima (value -1) at 3π/2 + 2kπ, and the tool marks them all as diamonds. The tangent, tan(x) = sin(x)/cos(x), has vertical asymptotes where cosine is zero, at π/2 + kπ, and period π instead of 2π.

Turn on x axis in multiples of π to read the labels π/2, π, 3π/2 and 2π instead of decimals like 1.571 and 3.142. Since waves are periodic, the properties panel estimates roots and extrema only over the region shown, and flags the result as an estimate. If you zoom out and see more periods, the list grows with it. Sine and cosine cross at π/4 + kπ, where both equal ±√2/2, about ±0.7071, and the intersections list confirms it.

Checked examples

Sine

Input: sin(x)
Expected output: f(0) = 0; f(π/2) = 1; f(π) = 0

Roots at multiples of π (seven between -10 and 10); maximum at (π/2, 1); odd function. Period 2π, amplitude 1.

Cosine

Input: cos(x)
Expected output: f(0) = 1; f(π/2) = 0

Y = 1 at x = 0; even function; first positive root at π/2. cos(x) = sin(x + π/2).

Amplitude 2

Input: 2*sin(x)
Expected output: [-2, 2]

Maximum 2 and minimum -2; same roots as sine. Amplitude |a| = 2.

Doubled frequency

Input: sin(2*x)
Expected output: T = π; f(π/2) = 0

Period π; roots at multiples of π/2. Period 2π/2 = π.

Shifted wave

Input: sin(x) + 1
Expected output: [0, 2]; f(0) = 1

Values from 0 to 2; y = 1 at x = 0. c = 1 raises the wave.

Sine and cosine together

Input: sin(x) e cos(x)
Expected output: (π/4, 0.7071)

First positive intersection at (π/4, 0.7071). Both equal √2/2.

Full tool FAQ

How do I graph a quadratic function?

How do I find the roots of a function on the graph?

How do I identify the asymptotes of a function?

What is the difference between an asymptote and a hole in a graph?

How do I plot several graphs and find where they cross?

What are the parameters a, b and c?

Which functions and symbols can I use?

Does the panel show the important points even off screen?

How do I save or share the graph?

Why does my function not show up or give an error?

Frequently asked questions