Reading and drawing graphs

A function as a picture

A formula tells you what to do with $x$. A graph shows you everything at once: where the function is high, where it is low, where it is zero and where it does not exist.

The graph of a function $f$ is the set of all points $(x, f(x))$ in the plane, with $x$ in the domain. To draw it by hand:

1. Find the domain, so you know where the graph can be.
2. Make a table of values, with a few $x$ and the matching $f(x)$.
3. Mark the points and join them with a smooth curve.
4. Check the special places: the zeros, the $y$-intercept and any value of $x$ that is not allowed.

A zero of $f$ is a value of $x$ with $f(x) = 0$. It is the place where the graph meets the $x$ axis. The $y$-intercept is $f(0)$, the place where the graph meets the $y$ axis.

What a graph tells you

• The value at a point. To find $f(3)$, go to $x = 3$ on the axis, go up or down to the curve, and read the height.
• The domain. It is the part of the $x$ axis that the curve covers.
• The range. It is the part of the $y$ axis that the curve covers.
• Where $f(x) > 0$. It is where the curve is above the $x$ axis.

Six graphs to know by heart

For $y = 1/x$, the curve comes closer and closer to the two axes but never touches them. A line that a curve approaches without meeting is called an asymptote. Here the asymptotes are the $x$ axis and the $y$ axis.

The parabola

The graph of $y = ax^2 + bx + c$, with $a \neq 0$, is a parabola. It opens upward when $a > 0$ and downward when $a < 0$. The picture below is $y = x^2 - 4x + 3$.

• Vertex. The lowest point (for $a > 0$) or the highest point (for $a < 0$). Its $x$ coordinate is $x = -\dfrac{b}{2a}$. Put this $x$ in the formula to get the $y$ coordinate.
• Axis of symmetry. The vertical line through the vertex. The two halves of the parabola are mirror images in it.
• Zeros. The solutions of $ax^2 + bx + c = 0$, which are $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. If $b^2 - 4ac > 0$ there are two zeros, if it is $0$ there is one, and if it is negative the parabola does not meet the $x$ axis.
• $y$-intercept. The value $c$.

For $y = x^2 - 4x + 3$: $a = 1$, $b = -4$, $c = 3$. The vertex is at $x = \dfrac{4}{2} = 2$, and $y = 4 - 8 + 3 = -1$. The zeros are $x = 1$ and $x = 3$, and the $y$-intercept is $3$. So the range is $[-1, \infty)$.

Completing the square shows the same thing in another way: $x^2 - 4x + 3 = (x - 2)^2 - 1$. A square is never negative, so the lowest value is $-1$, at $x = 2$.

Set $b = 0$. The vertex moves onto the y-axis. Change only $c$, and the parabola slides straight up or down.

Practise with new numbers

Try it in Python

This program draws the parabola and finds its vertex and zeros from $a$, $b$ and $c$.

Remember these

• The graph of $f$ is the set of points $(x, f(x))$.
• A zero is a value of $x$ with $f(x) = 0$. The $y$-intercept is $f(0)$.
• The vertex of $y = ax^2 + bx + c$ is at $x = -b/(2a)$.
• The graph of $y = 1/x$ has two branches, and the two axes are its asymptotes.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the vertex of $y = x^2 - 10x + 21$.

2. What is the range of $y = -x^2 + 4$?

3. What is the domain of $y = \dfrac{1}{x^2}$, and why is its range only $(0, \infty)$?

4. Where does $y = x^2 - 4x$ cross the x-axis?