Why a Calculator Graph Needs Checking
A graphing calculator turns a function into a picture, but the picture is not the function itself. The display has finite size, uses a chosen viewing window, and typically plots only selected points before drawing the visible curve. Those choices can make a real feature difficult to see or make separate pieces look connected.
When a graph seems to show a minimum, a zero, a vertical line, or a smooth curve through a gap, treat that appearance as a question to investigate. Check the function’s formula, its domain, and selected values in a table. Changing the window or zooming in can also help, but a clearer picture is not a substitute for checking the mathematics.
How the Window Changes What You See
A graphing window sets the displayed ranges of the horizontal and vertical axes. If the vertical range is very large, modest changes in function values occupy only a small part of the screen. A minimum may look flat, or two nearby values may appear nearly equal. With a narrower vertical range, the same differences may become visually prominent.
This effect follows from the way vertical distances are scaled. Suppose a display has \(H\) pixels from its lowest to highest vertical value, and the window shows \(y\)-values from \(y_{\min}\) to \(y_{\max}\). A change of \(\Delta y\) in function value corresponds to a screen distance of \(H\Delta y/(y_{\max}-y_{\min})\) pixels, in magnitude. The larger the vertical range, the fewer pixels represent a fixed change in \(y\).
This scaling explains why changing the window changes the appearance without changing the function. It also suggests a useful test: if an important feature is hard to see, adjust the window and compare the graph with a table of values.
Worked Example: A Minimum Hidden by a Wide Vertical Window
Consider \(f(x)=x^2\). Suppose a calculator display is \(400\) pixels high and its vertical window is \([-100,100]\), so the vertical range is \(200\).
At \(x=0\), \(f(0)=0^2=0\). At \(x=1\), \(f(1)=1^2=1\). The difference in function values is \(1-0=1\). The corresponding screen distance is \(400(1)/200=2\) pixels.
At \(x=2\), \(f(2)=2^2=4\). The difference from \(f(0)\) is \(4-0=4\), corresponding to \(400(4)/200=8\) pixels. These changes can be hard to distinguish on a display showing a vertical range of \(200\).
Now narrow the vertical window to \([-1,5]\), whose range is \(5-(-1)=6\). The change from \(f(0)=0\) to \(f(1)=1\) now corresponds to \(400(1)/6\approx66.7\) pixels. The minimum becomes much more visible.
Check: The function values remain \(f(0)=0\), \(f(1)=1\), and \(f(2)=4\) in both windows. Only their screen scale changes, so the minimum is real even if the first display makes it look nearly flat.
Check Suspicious Behavior Near an Asymptote
A graph may look as though it joins across a vertical asymptote if plotting points on opposite sides are connected, or if the display is too coarse to show the gap clearly. The formula can settle the issue. For a quotient, first identify where its denominator is zero; the function is not defined at those inputs. Then inspect values on both sides rather than relying on a line that appears to cross the excluded input.
Worked Example: Testing a Graph Near a Vertical Asymptote
Consider \(f(x)=1/(x-2)\). The denominator is zero when \(x-2=0\), so \(x=2\) is not in the domain. Calculate values on both sides:
At \(x=1.9\), \(x-2=-0.1\), so \(f(1.9)=1/(-0.1)=-10\).
At \(x=1.99\), \(x-2=-0.01\), so \(f(1.99)=1/(-0.01)=-100\).
At \(x=2.01\), \(x-2=0.01\), so \(f(2.01)=1/(0.01)=100\).
At \(x=2.1\), \(x-2=0.1\), so \(f(2.1)=1/(0.1)=10\).
The values become large in magnitude on both sides, negative to the left and positive to the right. At \(x=2\), the expression would require division by zero, so \(f(2)\) is undefined. The line \(x=2\) is a vertical asymptote: the function values grow without bound in magnitude as \(x\) approaches \(2\) from either side.
Check: Substituting \(x=2\) gives \(1/(2-2)=1/0\), which is undefined. Therefore, any apparent connection through \(x=2\) is not a point on the graph.
Rapid Oscillation Can Be Missed by Coarse Sampling
A display cannot plot every input value. If a function changes direction many times between the points the calculator samples, the drawn curve may hide oscillation. In particular, a set of sample points can all have the same output even when the function takes very different values between them.
Worked Example: Sampled Zeros Hide Oscillation
Consider \(f(x)=\sin(40\pi x)\), with the angle measured in radians. The sine function repeats after its input increases by \(2\pi\). Thus the period in \(x\) is \(2\pi/(40\pi)=1/20=0.05\).
A coarse table at \(x=0,\ 0.1,\ 0.2\) gives:
\(f(0)=\sin(0)=0\).
\(f(0.1)=\sin(40\pi(0.1))=\sin(4\pi)=0\).
\(f(0.2)=\sin(40\pi(0.2))=\sin(8\pi)=0\).
If only those points are considered, the function might appear to lie on the \(x\)-axis. But at \(x=0.0125\), the input to sine is \(40\pi(0.0125)=0.5\pi=\pi/2\), so \(f(0.0125)=1\). At \(x=0.0375\), the input is \(40\pi(0.0375)=1.5\pi=3\pi/2\), so \(f(0.0375)=-1\).
Check: The period is \(0.05\), while the coarse table spacing is \(0.1\), exactly two periods. Each listed sample therefore lands at the same phase and equals zero, even though the function reaches both \(1\) and \(-1\) between samples. A finer table or narrower viewing interval reveals the oscillation.
A Disconnected Domain Can Look Connected
The domain is the set of input values for which a function is defined. It must be checked before interpreting a plotted curve. For example, a square root of a real number requires its radicand to be nonnegative. If that condition produces separate intervals, there is no part of the real-valued graph between them, even if a coarse display seems to draw a bridge.
Worked Example: Finding a Gap in the Domain
Consider \(g(x)=\sqrt{x^2-1}\). For \(g(x)\) to be real, its radicand must satisfy \(x^2-1\geq0\). Factor the expression to get \((x-1)(x+1)\geq0\). This product is nonnegative when \(x\leq-1\) or \(x\geq1\), so the domain is \((-\infty,-1]\cup[1,\infty)\).
Some values in the domain are \(g(-2)=\sqrt{(-2)^2-1}=\sqrt{3}\), \(g(-1)=\sqrt{1-1}=0\), \(g(1)=\sqrt{1-1}=0\), and \(g(2)=\sqrt{2^2-1}=\sqrt{3}\).
At \(x=0\), the radicand is \(0^2-1=-1\), so \(g(0)=\sqrt{-1}\) is not real. In fact, no real value of \(g\) exists for any \(-1<x<1\).
Check: The domain calculation and the test at \(x=0\) confirm that the graph has no real-valued points in the gap. A displayed bridge across that interval cannot represent values of \(g\).
A Reliable Diagnostic Routine
When a graph looks surprising, use a sequence of checks. Each check answers a different question: Is the display hiding scale? Is the input allowed? Are values changing rapidly between samples? Does the formula support the apparent feature?
Describe exactly what looks suspicious: a crossing, a gap, a flat region, a spike, or a sudden change in direction.
Look for zero denominators, square-root restrictions, or other conditions that determine where the function is defined.
Evaluate at inputs on both sides of a suspicious point, and include the point itself if it is in the domain. Use closer inputs when behavior changes rapidly.
Zoom in or choose a more suitable vertical range. For rapid changes, inspect a smaller interval and use more closely spaced inputs.
Check whether the formula, domain, table, and revised graph tell a consistent story. Do not treat a visual connection as proof that the function is defined there.
Common Pitfall: Treating a Drawn Segment as a Function Value
Suppose the graph of \(f(x)=1/(x-2)\) appears to draw a segment between points on opposite sides of \(x=2\). A tempting but incorrect conclusion is that the segment passes through a value at \(x=2\), perhaps \(f(2)=0\), because it seems to cross the horizontal axis there.
That conclusion does not follow from the picture. Substitution gives \(f(2)=1/(2-2)=1/0\), which is undefined, not \(0\). The table also shows \(f(1.99)=-100\) and \(f(2.01)=100\), with opposite signs on opposite sides of the excluded input. The correction is to mark \(x=2\) as outside the domain and interpret the two sides separately; a plotted segment cannot supply a missing function value.
More generally, a graph may guide a useful question, but the formula and carefully chosen values are the checks that determine whether the apparent feature is consistent with the function. This matters in calculus because conclusions about zeros, limits, continuity, and behavior near a point depend on the function, not on how a finite display happens to connect pixels.
Check Your Understanding
For each question, explain what you would check before trusting the graph’s appearance.
- A curve appears almost flat near a minimum. How could the vertical window be responsible, and what table values would help?
- For \(h(x)=1/(x+3)\), what input is excluded from the domain, and what should you check on either side of it?
- Why can a coarse table of \(q(x)=\sin(20\pi x)\) miss changes that occur between its listed inputs?
- Find the real domain of \(r(x)=\sqrt{4-x^2}\). Which algebraic condition determines whether the graph can have a point at a given input?
- A graph appears to connect two branches across an interval where the formula is undefined. Which evidence should determine whether that connection is real?
