Curve sketching

Analysis mode takes a function of one variable and works through the whole curve sketch, each part with its working, beside an interactive graph. Letters are read as in Algebra.

Enter a function

Switch to Analysis in the top bar and enter the function in any of these forms:

f(x) = x^3 - 3x
y = x^3 - 3x
x^3 - 3x

Press Enter. Polynomials, rational functions, $e^x$, $a^x$, logarithms, roots, the trigonometric functions and their inverses and $|x|$ work, also combined. See Known limits for what is rejected.

The curve sketch

The Curve sketch tab lists one section per question. For $f(x) = x^3 - 3x$:

Section	Result
Derivatives	$f'(x) = 3x^{2} - 3, \quad f''(x) = 6x, \quad f'''(x) = 6$
Domain	$D = \mathbb{R}$
Symmetry	symmetric about the origin
y-intercept	$S_y(0 \mid 0)$
Zeros	$Z_{1}(-\sqrt{3} \mid 0), \; Z_{2}(0 \mid 0), \; Z_{3}(\sqrt{3} \mid 0)$
Extrema	$Max(-1 \mid 2), \; Min(1 \mid -2)$
Inflection points	$I(0 \mid 0)$
Monotonicity	increasing on $(-\infty, -1]$ and $[1, \infty)$, decreasing on $[-1, 1]$
Curvature	convex on $[0, \infty)$, concave on $(-\infty, 0]$
Behaviour at the edges and asymptotes	no asymptotes
Antiderivative	$F(x) = \frac{1}{4}x^{4} - \frac{3}{2}x^{2} + C$

Each section opens to its working. The zeros, for example:

Set $f(x) = 0$	$x^{3} - 3x = 0$
Factor out $x$	$x \cdot (x^{2} - 3) = 0$
A product is zero where one of its factors is zero	$x = 0 \lor x^{2} - 3 = 0$
Isolate the square	$x^{2} = 3$
Take the root	$x_{1,2} = \pm \sqrt{3}$

Results are exact where school methods reach, with a decimal next to them ($Z_{1}(-\sqrt{3} \mid 0) \approx Z_{1}(-1{,}73205 \mid 0)$). A section that needed a numeric method or ran out of time is marked PARTIAL.

Poles and asymptotes

For a rational function such as $f(x) = \frac{x^2+1}{x-1}$ the sketch also finds the gaps in the domain and the asymptotes:

Section	Result
Domain	$D = \mathbb{R} \setminus \{1\}$
Extrema	$Max(1 - \sqrt{2} \mid 2 - 2\sqrt{2}), \; Min(1 + \sqrt{2} \mid 2 + 2\sqrt{2})$
Behaviour at the edges and asymptotes	vertical asymptote $x = 1$, oblique asymptote $y = x + 1$
Antiderivative	$F(x) = \frac{1}{2}x^{2} + x + 2\ln\left\|x - 1\right\| + C$

The graph

The graph shows the curve, its asymptotes and the marked points (zeros, extrema, inflection and saddle points, removable gaps). Scroll to zoom, drag to move, Reset view to go back. Clicking a point selects it in the list below the graph, and the other way round.

Tangent and normal

On the Tangent / normal tab, enter $x_0$ and press Compute. For $x_0 = 2$:

Point on the graph	$f(2) = 2^{3} - 3 \cdot 2 = 2$
Slope: the derivative at $x_0$	$m = f'(2) = 3 \cdot 2^{2} - 3 = 9$
Tangent, values put in	$t(x) = 9 \cdot (x - 2) + 2$
	$t(x) = 9x - 16$
Normal: slope $-1/m$	$n(x) = -\frac{1}{9}x + \frac{20}{9}$

Both lines are drawn into the graph.

Area

On the Area tab, enter the bounds, or leave both empty for the area between the outermost zeros. Parts below the axis count with their absolute value; the signed INTEGRAL is shown too. For $x^3 - 3x$ without bounds:

Zeros inside the interval split it	$x = 0$
Integral over the part	$A_{1} = \left[F(x)\right]_{-\sqrt{3}}^{0} = \frac{9}{4}$
Integral over the part	$A_{2} = \left[F(x)\right]_{0}^{\sqrt{3}} = -\frac{9}{4}$
AREA	$A = \left\|A_{1}\right\| + \left\|A_{2}\right\| = \frac{9}{2}$
INTEGRAL	$\int_{-\sqrt{3}}^{\sqrt{3}} f(x)\,dx = 0$

Value table

On the Value table tab, enter from, to and step. Points outside the domain are shown as undef.

$x$	$-2$	$-1$	$0$	$1$	$2$
$f(x)$	$-2$	$2$	$0$	$-2$	$2$

Settings that matter here

• Quadratic equations: pq formula or abc formula in the working (Automatic: pq in German).
• Step detail: how much working each section shows.

See also

• Known limits