Why a Graph Can Give a Misleading Impression
A graph is a useful way to estimate what a function does near an input. But the picture is not the function itself: it is a display of selected input and output ranges, fitted into a finite rectangle and represented with a finite number of pixels. Changing any of these choices can change what the graph seems to show.
When estimating a limit from a graph, first identify the input value being approached and inspect the graph nearby, as in the earlier material on estimating limits from graphs. Then ask how the viewing window and plotting choices affect the evidence. A curve that looks nearly flat may simply have a very compressed vertical scale. A sharp change may appear gentle if the horizontal scale is compressed. A narrow feature may not appear at all if the plotting grid misses it.
How the Display Scales Coordinates
Suppose a graph displays inputs from \(a\) to \(b\) across a rectangle \(W\) pixels wide, and outputs from \(c\) to \(d\) across a rectangle \(H\) pixels high. One unit of input occupies \(W/(b-a)\) horizontal pixels. One unit of output occupies \(H/(d-c)\) vertical pixels.
For a small change \(\Delta x\) in input and the corresponding change \(\Delta y\) in output, the displayed vertical change divided by the displayed horizontal change is therefore
This ratio explains why the same function can look different in different graph windows. The factor \(\frac{H}{W}\frac{b-a}{d-c}\) comes from the display, not from the function. For a straight line with slope \(m\), the magnitude of its displayed slope is this factor multiplied by \(|m|\).
Why the Display-Scale Rule Works
To derive the rule, let \(\Delta x\) and \(\Delta y\) denote changes in the graph’s coordinates. The horizontal scale is \(W/(b-a)\) pixels per input unit, so the horizontal pixel change has magnitude \(\frac{W}{b-a}|\Delta x|\). Similarly, the vertical pixel change has magnitude \(\frac{H}{d-c}|\Delta y|\). Dividing the vertical change by the horizontal one gives
For a line, \(\left|\Delta y/\Delta x\right|=|m|\) for every pair of distinct points on it, which proves the stated rule. For a curve, the same calculation describes the displayed ratio between the two coordinate changes over a particular interval. The smaller the displayed distance from the target input, the more relevant that interval is to a local visual estimate.
Worked Example: The Same Line Looks Gentle or Steep
Worked Example: The Same Line Looks Gentle or Steep
Consider the line \(f(x)=2x\), shown in a graph that is \(600\) pixels wide and \(300\) pixels high. Compare two vertical windows while keeping the horizontal window \([-2,2]\).
First window: Let the vertical range be \([-4,4]\). The horizontal scale is \(600/4=150\) pixels per input unit. The vertical scale is \(300/8=37.5\) pixels per output unit. When the input increases by \(1\), the line’s output increases by \(2\), so its displayed rise is \(2(37.5)=75\) pixels. Its displayed run is \(150\) pixels. Thus its displayed slope magnitude is \(75/150=0.5\). The rule gives the same result:
Second window: Change the vertical range to \([-2,2]\), leaving the graph dimensions and horizontal window unchanged. The horizontal scale remains \(150\) pixels per input unit. The vertical scale is now \(300/4=75\) pixels per output unit. An input increase of \(1\) gives a displayed rise of \(2(75)=150\) pixels. The displayed run is still \(150\) pixels, so the displayed slope magnitude is \(150/150=1\). Using the rule:
The line looks twice as steep in the second window, even though its equation and coordinate slope have not changed. In that window, only the part of the line with outputs between \(-2\) and \(2\) is visible; the graph is clipped outside that range.
Check: In both windows, the change in output for an input increase of \(1\) is \(2\), exactly as \(f(x)=2x\) requires. The different displayed slopes come from the pixel scales, not from a change in the line.
Window Size Can Hide or Reveal Local Shape
A large window shows a broader part of a function but can squeeze small output changes into a few pixels. Narrowing the vertical range makes small output differences easier to see, provided the graph still includes the values of interest. Narrowing the horizontal range focuses attention near an input, but does not by itself guarantee that small vertical changes are visible.
Worked Example: A Curve That Looks Flat at One Scale
Consider \(f(x)=x^2\) near \(x=0\). First use horizontal and vertical ranges \([-1,1]\), displayed in a square \(400\)-by-\(400\)-pixel rectangle. Each vertical pixel represents \(2/400=0.005\) output units.
At \(x=0.1\), the function value is \(f(0.1)=(0.1)^2=0.01\). Its vertical distance above the \(x\)-axis is \(0.01/0.005=2\) pixels. At \(x=0.01\), the function value is \(f(0.01)=(0.01)^2=0.0001\), which is \(0.0001/0.005=0.02\) pixels above the axis. A displacement of \(0.02\) pixels is too small to be represented clearly as a separate vertical position, so the graph can look as if it lies on the axis there.
Now keep the horizontal range and pixel dimensions the same, but change the vertical range to \([-0.02,0.02]\). Each vertical pixel now represents \(0.04/400=0.0001\) output units. At \(x=0.01\), the value \(0.0001\) is one pixel above the axis. At \(x=0.1\), the value \(0.01\) is \(0.01/0.0001=100\) pixels above the axis.
Check: The computed function values are unchanged: \(f(0.1)=0.01\) and \(f(0.01)=0.0001\) in both displays. The first window places the latter value \(0.02\) pixels above the axis; the second places it \(1\) pixel above. The graph looks different because its vertical scale changed.
This example also illustrates why a broad view and a close view can be useful for different purposes. The broad view gives context; the close view gives more room to inspect small changes. Neither view should be mistaken for an exact record of every function value.
Aspect Ratio Changes the Picture
The aspect ratio describes the relative width and height of the graphing rectangle. Even with the same coordinate ranges, changing the rectangle’s proportions changes the displayed slope. Conversely, changing an axis range while keeping the rectangle fixed also changes how many pixels represent one coordinate unit. A line that appears close to horizontal or vertical is not, for that reason alone, evidence that its actual coordinate slope is small or large.
Worked Example: Recovering the Coordinate Slope
Worked Example: Recovering the Coordinate Slope
A line appears to rise \(1\) pixel for every \(2\) horizontal pixels in a graph. The graph is \(600\) pixels wide and \(300\) pixels high, shows inputs from \(-2\) to \(2\), and shows outputs from \(-4\) to \(4\). Find the line’s coordinate slope.
The displayed slope magnitude is \(1/2\). The scale factor is
By the display-scale rule, \(\frac12=|m|\cdot\frac14\). Therefore \(|m|=(1/2)/(1/4)=2\). If the line rises from left to right, its coordinate slope is \(m=2\).
Check: A coordinate slope of \(2\) displayed with scale factor \(1/4\) gives \(2(1/4)=1/2\), matching the observed rise-to-run ratio in pixels.
Plotting Resolution Can Miss a Feature
Many plotting programs evaluate a function at a finite collection of input values, then draw between the resulting points. If the input samples are too far apart, the graph can miss a narrow feature between them. Zooming the axes may not solve this problem unless the plotting program also uses enough sample points in the newly visible region.
Worked Example: A Narrow Peak Between Sampled Inputs
Define a function by
The added term is positive only when \(1-100|x-0.45|>0\), which is equivalent to \(|x-0.45|<0.01\). Thus the feature is confined to \(0.44<x<0.46\). Suppose the graphing program samples only at inputs \(x=k/10\), where \(k\) is an integer. The nearest sampled inputs to \(0.45\) are \(0.4\) and \(0.5\).
At \(x=0.4\), the added term is \(\max(1-100|0.4-0.45|,0)=\max(1-5,0)=0\), so \(f(0.4)=0.4\). At \(x=0.5\), it is \(\max(1-100|0.5-0.45|,0)=\max(1-5,0)=0\), so \(f(0.5)=0.5\). In fact, every sample at a multiple of \(0.1\) has the added term equal to zero. Connecting those sampled points would draw the same straight segment as \(y=x\) between \(0.4\) and \(0.5\).
But at the unsampled input \(x=0.45\), the added term is \(\max(1-100(0),0)=1\), so \(f(0.45)=0.45+1=1.45\). The coarse plot has missed the peak. A finer sampling grid with spacing \(0.001\) includes \(0.45\), and also samples within the feature. For example, at \(x=0.449\), the added term is \(1-100(0.001)=0.9\), giving \(f(0.449)=1.349\); at \(x=0.451\), it is also \(0.9\), giving \(f(0.451)=1.351\).
Check: The added term vanishes at both coarse sample points, so their plotted values are \(0.4\) and \(0.5\). At the feature’s center it equals \(1\), so the actual function value is \(1.45\). The coarse plot’s straight segment cannot show that peak because none of its sample points record it.
A Common Mistake: Reading Pixel Slope as Coordinate Slope
A common error is to treat the slope measured on the screen as the slope in the function’s coordinates. In the line example above, the screen ratio was \(1/2\), so the incorrect calculation would be “the line’s slope is \(1/2\).” That ignores the scale factor \(1/4\). The correction is to divide the displayed slope by the scale factor: \((1/2)/(1/4)=2\). Always check the axis ranges and the display dimensions before interpreting a line’s visual steepness.
Another caution is that a plot with many pixels is not automatically a plot with many function evaluations. Pixel resolution describes how finely the image can represent positions; sampling resolution describes how many input values the program checks. A high-resolution image can still omit a feature if its input samples are too widely spaced.
A Practical Way to Inspect a Graph
When a graph seems to suggest a particular local trend, use the following checks before trusting the visual impression:
Read the horizontal axis carefully and identify the input near which the behavior is being estimated.
Note the displayed horizontal and vertical intervals. A narrow vertical interval magnifies small output changes; a broad one compresses them.
Remember that a graph rectangle’s width and height affect the displayed slope, even when the coordinate ranges are unchanged.
Ask whether the graph may be joining widely spaced samples or rounding small changes to the same pixel row.
Use a broad window for context and a closer window for local detail. If the apparent behavior changes, inspect the scale settings before drawing a conclusion.
These checks do not turn a graph into a proof. They help distinguish what the function may be doing from what a particular display can show. A responsible visual estimate accounts for the window, aspect ratio, and resolution, and treats the picture as evidence rather than as a guarantee.
Check Your Understanding
Use the graph-scale ideas in this tutorial to answer the following questions.
- A graph’s vertical range is narrowed while its horizontal range and pixel dimensions remain fixed. What happens to the displayed vertical distance corresponding to a fixed output change?
- Why can a line’s displayed slope differ from its coordinate slope?
- For \(f(x)=x^2\), explain why \(f(0.01)\) may be visually indistinguishable from zero in a window where each vertical pixel represents \(0.005\) output units.
- What two different kinds of resolution should you consider when a narrow feature is absent from a graph?
- In the narrow-peak example, why do samples at multiples of \(0.1\) fail to reveal the feature centered at \(0.45\)?
