Plotting and Reading Graphs

A graph is the function's autobiography. Every input and its output appear together as a point in the plane, and the shape of the resulting curve broadcasts every important property: where the function vanishes, where it peaks, whether it is even or odd, what it does far from the origin. Learning to read graphs is the difference between manipulating symbols and actually understanding the function.

The graph as a point set

The graph of a function $f$ is the set of points

$\{(x, f(x)) : x \in \text{domain}(f)\} \subset \mathbb{R}^2.$

Plotting that set on coordinate axes produces a curve. The horizontal coordinate tells you the input; the height of the curve above (or below) it tells you the output.

The vertical line test

A curve in the plane is the graph of a function if and only if every vertical line intersects it in at most one point. This is just the definition restated visually: each input $x$ has at most one output, so the vertical line $x = a$ can hit the curve at most once. The circle $x^2 + y^2 = 4$ fails the test (at $x = 0$ both $y = 2$ and $y = -2$ lie on it), so it is not a function graph.

Standard library

Six graphs to memorise, they recur everywhere:

• $y = x$, line through origin at 45^\circ.
• $y = x^2$, parabola opening up, vertex at origin.
• $y = x^3$, S-shaped cubic, passing through origin with a flat moment.
• $y = |x|$, V opening up.
• $y = \sqrt{x}$, right half of a sideways parabola.
• $y = 1/x$, hyperbola with two branches, asymptotes at both axes.

Transformations, shift, reflect, stretch

Starting from $y = f(x)$, the transformations follow a simple grammar:

• $y = f(x) + c$ shifts the graph up by $c$.
• $y = f(x - c)$ shifts the graph right by $c$.
• $y = -f(x)$ reflects across the $x$-axis.
• $y = f(-x)$ reflects across the $y$-axis.
• $y = a \, f(x)$ stretches vertically by $|a|$.

The shifts look reversed from what you might expect, $f(x - c)$ moves the graph to the right, not left, because the formula asks for an $x$-value $c$ units bigger to produce the same output.

Symmetry: even and odd

A function is even if $f(-x) = f(x)$ for every $x$; its graph is symmetric about the $y$-axis. It is odd if $f(-x) = -f(x)$; its graph is symmetric about the origin (rotate the plane 180^\circ and the curve maps to itself). The polynomial $x^4 - x^2$ is even; $x^3 + x$ is odd. Most functions are neither.

• Data Visualisation: Every line chart, bar chart, and scatter plot is a function graph; identifying intercepts (when did sales hit zero?) and extrema (when was the peak?) is the primary language of business analytics.
• Physics: Phase diagrams and field maps are 2D function graphs; the contour at $f = 0$ is the zero-energy locus, and extrema correspond to equilibrium states.
• Image Processing: Histogram-equalisation transformations on an image apply a specific function to each pixel intensity; the function's graph is exactly the lookup-table the photo editor uses to brighten or darken.

(Pick a family and drag the coefficients. Note how every transformation in the list above appears as a slider: $h$ shifts horizontally, $k$ shifts vertically, the sign of $a$ reflects.)

Try it

• Predict first: for $y = a|x|$ with $a = -1$, which way should the V open? Pick the absolute-value family and drag $a$ to $-1$; explain in one sentence why a negative $a$ flips the V.
• Before setting: for $y = -x^2 + 1$, does the parabola open up or down, and where is its axis of symmetry? Set $a = -1, b = 0, c = 1$ and verify it opens downward and is symmetric about the $y$-axis.
• Sketch on paper the transformation from $y = x^2$ to $y = -(x - 3)^2 + 1$. Verify in the widget.

Pause: a function passes the vertical line test. Could its graph still be a closed curve? Why or why not?

A trap to watch for

Beginners read $y = f(x - 3)$ and shift the graph left by 3 because the formula has a minus sign. This is backwards. The shift goes in the opposite direction from what the sign suggests because $f(x - 3)$ is asking the original $f$ to accept a smaller input than $x$, specifically, three less. To get the same output as the old $f$ produced at, say, $x = 0$, you now need to feed in $x = 3$. So the curve shifts to the right. Rule of thumb: solve for what $x$ makes the inner expression zero, and that is where the old origin lands.

What you now know

You can plot a function as a point set, decide whether a curve is a function graph (vertical line test), recognise the six standard graphs by silhouette alone, apply shifts/reflections/stretches by reading them off the formula, and test for even/odd symmetry both algebraically and visually. The next two sections introduce two profoundly important non-polynomial families, the exponential and the logarithm, whose graphs you will recognise instantly once you have spent five minutes with the widget.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 13 §3, graphs of polynomial functions, transformations.
• Stewart, J. (2015). Calculus: Early Transcendentals, 8th ed. Cengage. Sections 1.2–1.3, the standard library, transformations, and symmetry.
• Spivak, M. (2008). Calculus, 4th ed. Publish or Perish. Chapter 4: graphs as point sets and the vertical line test.