Course 1.2Lesson 2 of 5

Graphs of functions

To draw the graph of a function, make an array of x values with np.linspace, calculate y from the formula for the whole array, and plot the two. More points give a smoother curve, because plt.plot joins the points with straight pieces, so a curve needs about 100 or more. Several curves can share one graph with labels and a legend. Angles for sine and cosine are changed to radians with np.deg2rad. plt.xlim and plt.ylim set the range, plt.axhline and plt.axvline draw the axes, plt.xticks chooses the ticks, plt.plot with a marker shows special points, and plt.annotate writes a note at a point. The meeting points of two graphs are the solutions of an equation, and np.roots finds the roots of a polynomial.

about 35 min1 simulation2 practice questions

Lesson 2 of 5Show all lessons

Try the code and the simulations beside the lesson.

What you will be able to do

After this lesson you can

draw the graph of any formula, such as a parabola or a sine wave,
choose enough points for a smooth curve,
put several functions on one graph and control the range and the ticks, and
mark and name special points, such as the place where two graphs meet.

A function is an array and a formula

In your maths class, to draw $y = x^2$ you made a table of values of $x$ and $y$ , plotted the points, and joined them. With NumPy, you do the same, but with hundreds of points and no effort: you make an array of $x$ values, calculate the array of $y$ values with the formula, and plot them.

Predict first. Which line will look smoother, the one with 200 points or the one with 5 points?

The graph of y equal to x squared from minus 3 to 3. A smooth blue curve is drawn with 200 points and an orange dashed line with five round markers is drawn with 5 points. The orange line has corners and is a rough copy of the smooth curve.

np.linspace(-3, 3, 200) gives 200 values of $x$ from $-3$ to $3$ . And many ** 2 is the 200 values of $y$ . (We use ** for the power.)
plt.plot joins the neighbouring points with straight lines. With 5 points, the picture has corners, because it is made of 4 straight pieces. With 200 points, the pieces are so short that the eye sees a smooth curve.
The rule of thumb: use at least 100 points for a curve. More points cost almost nothing.

Notice the title: "y = x squared". In a title, you cannot easily write the power, so it is written in words, or with a short form such as x^2 as a text. The title is only a text.

Sine and cosine

A graph of sine in blue and cosine in orange for angles from 0 to 360 degrees, with a legend, a grid and ticks every 90 degrees. Sine starts at 0, rises to 1 at 90 degrees, is 0 at 180, is minus 1 at 270 and is 0 at 360. Cosine starts at 1, is 0 at 90, minus 1 at 180, 0 at 270 and 1 at 360.

For a smooth graph, 100 to 200 points are plenty. They cost nothing, because NumPy works them out in one line.

Where do two graphs meet?

The places where two graphs meet are the solutions of an equation. The parabola $y = x^2 - 4$ and the line $y = 2x - 1$ meet where $x^2 - 4 = 2x - 1$ . Move everything to one side: $x^2 - 2x - 3 = 0$ .

Predict first. Solve $x^2 - 2x - 3 = 0$ by factorising: $(x - 3)(x + 1) = 0$ . Where do the graphs meet? Then run it.

A parabola, y equal to x squared minus 4, and a straight line, y equal to 2x minus 1, on a grid with the axes drawn in black. Two red dots mark where they meet, at minus 1 and minus 3, and at 3 and 5, and each is labelled with its coordinates.

np.roots([1, -2, -3]) gives the roots of the polynomial $1x^2 - 2x - 3$ : the coefficients are listed from the highest power. The answer is the two numbers, 3 and -1, as you found by factorising. (Chapter 3 returns to this function. Rounding to 6 decimals removes the tiny errors in the last digits.)
plt.plot(roots, 2 * roots - 1, "ro") draws two red dots (r is red, o is a round marker) at the meeting points. A dot is a plot of its own, with only markers.
plt.annotate(text, (x, y), ...) writes a note near a point. xytext=(8, -14) moves the text 8 points to the right and 14 points down from the point, so that the note does not sit on the line.
plt.xlim(-4, 5) and plt.ylim(-8, 14) set the range that is shown. Without them, matplotlib picks a range. A good range shows the important part of the graph: here, both meeting points and the shape of the parabola.

The graph and the equation say the same thing in two ways. The graph shows where the solutions are. The calculation gives them exactly.

Common mistakes

Too few points. A "curve" with corners.
Degrees given to np.sin. Use np.deg2rad first.
Forgetting the legend when two curves share a graph.
A range that hides the interesting part. Set xlim and ylim.
Writing y = x^2 in a calculation. In Python, the power is **.
Putting the roots in the wrong order in np.roots. The coefficients start with the highest power.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

Warm-up. Open the lab Plot a function. You write a function that draws any formula.
Build. Open the problem A parabola and its roots and the problem Waves of different size.

Remember these

To draw a function: x = np.linspace(...), y = formula(x), plt.plot(x, y).
Use at least 100 points for a smooth curve.
np.deg2rad before np.sin and np.cos.
plt.xlim, plt.ylim, plt.axhline, plt.xticks and plt.annotate control and explain the picture.
Where two graphs meet, the equation of both is true. np.roots solves a polynomial equation.