Graphing calculator

How to graph a function

Write the function in terms of x, such as y = x² − 4, typed as x^2 - 4. Its graph is every point (x, f(x)). This parabola crosses the x-axis at x = −2 and x = 2, where f(x) = 0, and its lowest point is the vertex (0, −4).

To sketch a graph by hand, or to check a plotted one, work out a few points and look for the features that matter: where the curve crosses the axes, where it turns, and where it is undefined.

Check a graph with a table of values

Substituting x values gives points the graph must pass through. For the parabola, f(−2) = (−2)² − 4 = 0. The cubic x³ − 3x is odd, so f(−x) = −f(x): its values at −1 and 1 are 2 and −2. The line 2x + 1 has slope 2 and crosses the y-axis at 1.

A table checks a few coordinates. It cannot show a narrow peak, closely spaced roots or a break that falls between the sampled points, so substitute a candidate back into the expression when an exact answer matters.

How to type functions

• Powers and products: use ^ for powers and * for multiplication, as in x^3 - 3*x or x*sin(x). A number written directly before x, as in 2x, also multiplies.
• Grouping: 1/(x-1) is one over x − 1, while 1/x-1 is one over x, minus one.
• Angles: trigonometric functions take radians, so sin(pi/2) = 1 and sin(90) is the sine of 90 radians. For degrees, graph sin(pi*x/180), or use the scientific calculator in degree mode for single values.
• Logarithms: log(x) is base 10 and ln(x) is base e. log2(x) gives base 2.
• Polar and parametric curves: start with r = for a polar curve, as in r = 1 + cos(θ), or type a pair such as (cos(t), sin(t)).

Graphs of sin x, cos x and tan x

sin(x) and cos(x) are waves between −1 and 1 that repeat every 2π, about 6.2832. Sine starts at 0 and first peaks at x = π/2 ≈ 1.5708; cosine starts at its peak, 1, and is the same wave shifted left by π/2.

tan(x) = sin(x)/cos(x) repeats every π and is undefined wherever cos(x) = 0, at x = π/2 + kπ for any integer k. Near those vertical asymptotes the curve heads to +∞ on one side and −∞ on the other.

OpenStax's graphs of the other trigonometric functions explains tangent's period and asymptotes in detail.

Multiplying by x changes the shape. xsin(x) swings between the lines y = x and y = −x and no longer repeats, while xtan(x) keeps the asymptotes of tan(x). Both graphs are symmetric about the y-axis.

Graphs of ln x and log x

ln(x) and log(x) are defined only for x > 0, so both graphs lie to the right of the y-axis. Both pass through (1, 0), keep rising, and fall toward −∞ as x approaches 0. ln(x) reaches 1 at x = e ≈ 2.718, and log(x) reaches 1 at x = 10.

Because log(x) = ln(x)/ln 10, the base-10 curve is the natural-log curve squashed vertically by a factor of about 2.3026. OpenStax covers the shapes in graphs of logarithmic functions.

The products xln(x) and xlog(x) have the same domain, x > 0. They approach 0 as x approaches 0 but are not defined at 0 itself. x*ln(x) dips to its minimum, −1/e ≈ −0.3679, at x = 1/e.

A sine or cosine function with a given period

For sin(Bx) or cos(Bx) with x in radians, the period is 2π/|B|, so a period P needs B = 2π/P (OpenStax):

• A sine function with period 2/3 is sin(3pix), because 2π/(3π) = 2/3.
• A cosine function with period 7 is cos(2pix/7), because 2π/(2π/7) = 7.

Both have amplitude 1, no vertical shift and no phase shift. Changing any of those gives another function with the same period.

These values at x = 1 and x = 2, rounded to eight significant digits, check the curves above:

Roots and intersections

A root is where f(x) = 0, so the graph meets the x-axis. A root where the curve only touches the axis, as y = x² does at 0, has no sign change on either side and is easy to miss numerically; zoom in to confirm it.

Two graphs intersect where f(x) = g(x). The curves y = x² and y = 2x cross at (0, 0) and (2, 4), the solutions of x² − 2x = 0. A read-off coordinate is rounded, so substitute it back when precision matters.

For exact roots of a quadratic, use the quadratic equation solver; for lines, distances and midpoints, the coordinate geometry calculator.

Slopes and areas under a curve

The derivative is the slope of the graph: positive where the curve rises, zero at a turning point and negative where it falls. The area under a curve between two limits is the definite integral, which counts regions below the x-axis as negative.

For y = x² from 0 to 1 the area is exactly 1/3. Reversing the limits flips the sign, and a curve that breaks or is undefined inside the interval may have no finite area there.