Graphs & functions / Matrices, vectors & complex — formulas and numbers as graphs, lines and arrows

The "x" card at the bottom left is an input card; its slider runs from −2 to 2 in steps of 0.1 (see Input types). The "x²" card at the top is a function card. It plots f(x) = x^2 and marks the point at the current x. "f(0.5) = 0.25" inside the plot is the value at that point. The number line card on the right places the two values, x and x², on one line.

With x at 1.5, the plot shows f(1.5) = 2.25 and x² moves to the right of x on the number line. With x at 1, both values are 1 and the two points sit at the same place. Squaring a number between 0 and 1 gives a smaller number. The number line picks its scale again each time, to fit the values it shows.

Function

Given a formula in x, this card plots its graph. The formula goes in the "f(x) =" field, for example x^2 or a*x+b. It is written the same way as in a calc card (Formulas and functions), and a product such as 2x can be written without the multiplication sign. The two "x:" fields set the range of the horizontal axis; a new card starts at −10 to 10. The vertical axis fits itself to the curve.

Letters other than x in the formula (such as a or b) take their values from connected cards with the same name. The names must be letters only, such as a or mu; a name with a digit, such as a1, is not used. A connected card named "x" puts a point on the curve at that x and shows the value, as in "f(2) = 4". That value is passed on to other cards. If the formula has a letter with no connected card, or cannot be read, nothing is plotted and the card shows "Enter a function of x". With sin, cos or tan in the formula, the card shows "Angles: degrees (x is in degrees)" next to the range fields (the angle unit is explained in the trigonometry part of Formulas and functions).

"Examine the graph ▾" opens four more fields. A field with a value in it stays visible after the section is closed. The plot gets smaller as fields open; dragging the card's bottom-right corner makes the card larger.

• vs — a second formula, drawn as a dashed curve on the same axes.
• Tangent x₀ — given an x value, the card draws the tangent line there and shows its slope, as in "f′(1) = 2". When the derivative can be worked out from the formula, it is shown too, as in "f′(x) = 2*a*x + b".
• Area ∫ — given the start and end of an interval, the card shows the definite integral over it and shades the region between the curve and the horizontal axis. Parts below the axis count as negative. Choosing n=4 to n=32 in the list on the right adds the value from n rectangles.
• Zeros, extrema & crossings — when ticked, the card marks where the curve meets the horizontal axis (zeros), its peaks (max) and valleys (min), and where it crosses the "vs" curve, with their coordinates.

Board 2 changes a, b and c of f(x) = a·x² + b·x + c with three input cards shown as sliders. With "Examine the graph ▾" open and "Zeros, extrema & crossings" ticked, the plot shows zeros at x = −1.414, 1.414 and a min at (0, −2). With c at 1, the parabola no longer meets the axis: the zeros read "none" and the min is (0, 1). With c back at −2 and b at 2, the zeros move to x = −2.732, 0.732 and the min to (−1, −3). With b back at 0 and a at −1, the parabola opens downward and a max at (0, −2) replaces the min.

With a back at 1 (b is 0, c is −2), entering 1 in "Tangent x₀" shows f′(x) = 2*a*x + b and f′(1) = 2. Setting "Area ∫" to 0 to 2 gives ∫[0, 2] = −1.333; over this interval, most of the curve lies below the axis.

Three more boards use the function card. "Move a line" changes a and b of f(x) = a·x + b. Its "vs" field holds a*x, so the gap between the solid and the dashed line is b. "A point on the curve" connects a card named "x" to f(x) = a·x². With x at −3 it shows f(−3) = 9, and with a then at −1, f(−3) = −9. "Repeated growth" takes 100 and increases it by r% n times: a calc card gives the value and a function card draws the whole curve. With r at 5 and n at 10 the value is 162.889463; with r at 10 it is 259.374246, and with n then at 20, 672.749995.

Open the line board ↗

Open the point-on-a-curve board ↗

Open the repeated-growth board ↗

Normal distribution

"Normal distribution" places three cards at once: input cards for the mean, "mu" (0), and the standard deviation, "sigma" (1), and a function card with the normal density exp(-(x-mu)^2/(2*sigma^2))/(sigma*sqrt(2*pi)). The function card's range is −4 to 4, and its "Area ∫" is already set to −1 to 1, showing ∫[-1, 1] = 0.683: about 68% of the whole lies within one standard deviation of the mean. The third card is an ordinary function card, so everything in "Function" above applies to it.

The board below is a score distribution with mean 50 and standard deviation 10. The area from 40 to 60 is 0.683. With the area interval at 60 to 80 it is 0.157, and at 30 to 70 it is 0.954.

Open the score distribution board ↗

Unit circle

Given an angle θ in degrees, this card draws the point on the unit circle (radius 1) and the segments for sin, cos and tan. On the right it draws the wave of the chosen function from 0° to 360°, with a mark at θ. θ can be typed into its field or set by dragging on the circle (dragging moves in whole degrees). Angles are brought into 0° to 360°: −30 becomes 330°, and 400 becomes 40°.

The value picked in the list at the top right ("sin" at first) is the card's large number and is passed on to other cards. Below it are θ in radians and the values of the other two functions. When the value is undefined, as for tan 90°, the card shows "—" and a card that uses it shows "No value to reference". The angle cannot come from a wire; connecting one shows "This card can't take a value". The board below has θ at 150°: "sin 150° = 0.5", 5π/6 in radians, and cos −0.866.

Open the unit circle at θ = 150° ↗

Number line

Given connected cards that show numbers, this card places their values as points on one number line. Each point is labelled with its value and the connected card's name; a name longer than seven characters is shortened to six and "…". It places up to eight cards, in the order they were connected. The scale fits the values, and includes 0 when the values are close to 0. With nothing connected, it shows "Connect value cards". The number line card does not pass a value on.

Connecting one card that holds a group of numbers, such as "Current & goal" or "Min / current / max", places the numbers in it with labels such as "Current" and "Goal". For "Min / current / max", the stretch from min to max is shaded as a band. "Number line" can also appear among the suggestions when several number cards are selected and "Visualize" is pressed (Visualize). Board 1 above is an example.

Log number line

This is the number line card placed with "Log scale" ticked. The scale goes 1, 10, 100, each step ten times the one before, so numbers of very different sizes fit on one line. Unticking the box turns it back into an ordinary number line. Tick labels of 100,000 and above are written as 1e5 and so on; 1e5 means 10 to the power 5.

Values of 0 or below cannot be placed on this scale. The card then shows how many were left out, as in "1 value(s) at or below 0 cannot be placed", and places the rest. When every value is 0 or below, it shows "A log scale needs values above 0".

Board 3 places four lengths in millimetres, from the width of a hair, 0.1 mm, to the height of Mt Fuji, 3,776,000 mm (3,776 m). The scale runs from 0.1 to 1e7, and the four points sit apart on one line. With "Log scale" unticked, the scale runs from 0 to 4,000,000, and the three values other than Mt Fuji pile up at 0.

Matrix

The cards from here on are in "Matrices, vectors & complex". Each of them takes its numbers in its own fields.

"Matrix" holds one matrix. The − and ＋ buttons for "Rows" and "Cols" set its size from 1 to 6. Cells take whole numbers, decimals and fractions such as 1/2. For a square matrix the card shows the "Determinant", and that value is passed on to other cards. It shows "Not a square matrix" when the matrix is not square, "Some cells are empty. Fill in every cell" when a cell is empty, and "Some cells are not numbers" when a cell cannot be read as a number.

In the board below, a calc card takes the determinant of the 2 × 2 matrix A (first row 2 and 1, second row 1 and 3) as abs(A). The parallelogram with sides (2, 1) and (1, 3) has an area equal to the absolute value of this determinant, 5.

Open the determinant board ↗

Matrix calc

This card calculates with two matrices, A and B. The list offers "Add (+)", "Subtract (−)", "Multiply (×)", "Scale by a number", "Transpose", "Inverse" and "Solve the system". The − and ＋ buttons for "ARows", "ACols", "BRows" and "BCols" set the sizes; once the cells are filled, the answer appears on the right. Fractions stay fractions: the inverse of the matrix with first row 2 and 1 and second row 1 and 3 is 3/5, −1/5, −1/5, 2/5.

Shapes that do not fit show "The shapes do not match", and a matrix with no inverse shows "No single answer (no inverse)". "Solve the system" treats A as the coefficients and B as the right-hand side, and gives the unknowns. The "System of 3 (matrix)" tile in "Equations & formulas" places this card set up for three unknowns (Equations & formulas). The matrix calc card does not pass a value on.

Board 4 totals two shopping baskets with a matrix product. Each row of A is one basket: the first has 3 and 5 of two items, the second 2 and 4. B holds the two unit prices, 120 and 80. The answers, 760 and 560, are the two totals. Selecting an answer cell shows its calculation below, as in "3×120 + 5×80 ＝ 760", and highlights the row and column it uses.

"Solve it myself" empties the answer cells so you can enter your own answers. Correct cells get a green border and wrong ones a red border, and the card shows how many are right, as in "1/2 correct". "Show the answer" brings the answer back.

Vector

This card calculates with two vectors in the plane, a and b. The list offers "Add (a + b)", "Subtract (a − b)", "Scale (k a)" and "Dot product (a · b)", and the components go in the fields. Components can be whole numbers, decimals or fractions; decimals are turned into fractions, so 1.5 shows as 3/2. The card shows the calculation component by component and the length of the resulting vector, with an arrow diagram below. A new card adds a = (3, 1) and b = (1, 2): a + b = (4, 3), with length |a + b| = √25 = 5.

The value passed on is the length of the resulting vector for add, subtract and scale, and the dot product itself for the dot product. For the dot product the card also shows cos θ and the angle between the vectors, as in "Angle θ ≈ 45°". With a zero vector it shows "The angle is undefined (zero vector)". The board below takes the dot product of a = (2, 1) and b = (1, 3): a · b = 2 × 1 + 1 × 3 = 5, and the angle is about 45°.

Open the dot product board ↗

Complex number

This card calculates with two complex numbers, z₁ and z₂. The list offers "Add (+)", "Subtract (−)", "Multiply (×)" and "Divide (÷)", and the real and imaginary parts go in the fields. The fields take whole numbers and decimals; a fraction such as 1/2 gives "Some number cannot be read (e.g. 3, −1.5, 0.25)". The card shows the working, with a short note such as "since i² = −1" for multiplication or "multiply top and bottom by (1 − i)" for division, then the length and angle of the answer, and arrows on the complex plane.

For multiplication and division, the card also shows that the lengths multiply (divide) and the angles add (subtract). The value passed on is the length (absolute value) of the answer. Dividing by 0 shows "Cannot divide by 0". A new card multiplies (3 + 2i)(1 + i), which gives 1 + 5i. The board below multiplies (2 + i) by i, which gives −1 + 2i. It shows "length 2.24 × 1 = 2.24, angle 26.57° + 90° = 116.57°": the length stays the same and the arrow turns 90° counterclockwise.

Open the multiply-by-i board ↗

Read next — Shapes, units & dates Cards for shapes, unit conversion, and dates and durations.

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