What It Means for Two Graphs to Intersect
Two graphs intersect when they share a point. If the graphs are \(y=f(x)\) and \(y=g(x)\), a shared point must have the same \(x\)-coordinate on both graphs and the same \(y\)-coordinate. Thus an intersection coordinate \((a,b)\) satisfies both \(b=f(a)\) and \(b=g(a)\).
A graphing calculator can help locate possible intersections, especially when the equation \(f(x)=g(x)\) is difficult to solve exactly. But the calculator’s displayed coordinates are numerical approximations. A graph that looks as if it meets another graph is not, by itself, enough evidence: the graphs might only pass very close to one another. A reliable method combines a graph or solver with a check in the original functions.
Turn the Intersection into an Equation
Define the difference function \(h(x)=f(x)-g(x)\). Then the two graphs intersect at an input \(x=a\) exactly when \(h(a)=0\). This reformulates the task as finding a zero of one function. It also provides a convenient check: after a calculator reports a candidate \(\hat{x}\), evaluate \(h(\hat{x})\), or evaluate \(f(\hat{x})\) and \(g(\hat{x})\) separately and compare the results.
The equivalence follows directly from subtracting \(g(a)\) from both sides of \(f(a)=g(a)\), or adding \(g(a)\) to both sides of \(f(a)-g(a)=0\). It explains why both a calculator’s intersection feature and a root-solving feature can be useful: the first seeks common points on the two graphs, while the second can solve the difference equation.
Finding a zero numerically does not establish that the calculator has found every zero, or even that a displayed approximation is an exact zero. The answer should be treated as a candidate until it is checked. The graph window, chosen starting value, and rounding can all affect what is found or displayed.
A Reliable Calculator Workflow
Check parentheses, signs, and the calculator’s angle mode if the functions involve trigonometric expressions. A mistyped function produces a consistent-looking but irrelevant result.
Use the stated domain or the interval relevant to the question. Adjust the window so both graphs and their possible meeting points are visible.
Use the calculator’s intersection tool, or solve \(f(x)-g(x)=0\). If there may be several intersections, search in separate regions rather than accepting the first one found.
Substitute the candidate \(x\)-value into \(f\) and \(g\) separately. Their outputs should agree to the precision being reported.
Give the point as \((x,y)\), state that it is approximate, and retain enough digits to make the verification meaningful.
When using a root-solving tool, solve for the zero of \(h(x)=f(x)-g(x)\), not for the zero of either original function unless that is what the problem asks. For example, \(f(x)=0\) identifies where the first graph crosses the \(x\)-axis; it does not generally identify where \(f\) and \(g\) intersect.
Worked Example: An Exponential Graph and a Line
Worked Example: An Exponential Graph and a Line
Find the intersection of \(y=e^x\) and \(y=3-x\) numerically.
Step 1: Form the difference equation. Set the two function values equal and move everything to one side:
Step 2: Locate a candidate. Enter \(Y_1=e^x\) and \(Y_2=3-x\), then use the calculator’s intersection feature near the visible meeting point. Equivalently, use a numerical root feature on \(h(x)=e^x+x-3\). The calculator gives \(x\approx0.7920599684\).
Step 3: Check both \(y\)-values. Using the candidate to the displayed precision, the exponential gives \(e^{0.7920599684}\approx2.2079400316\). The line gives \(3-0.7920599684=2.2079400316\) to the shown precision. The two outputs agree at this precision.
Step 4: State the point. The numerical intersection is approximately \((0.7920599684,\,2.2079400316)\). If only six decimal places are needed, report \((0.792060,\,2.207940)\).
Check: Substituting the reported \(x\)-coordinate into both original functions gives the same \(y\)-coordinate to the stated precision. This is the relevant check; checking only that the point looks like it lies on one graph would not be sufficient.
Multiple Intersections Require Multiple Checks
Two graphs can intersect more than once. A graphing calculator’s intersection feature may return a point near the cursor or within the current viewing region; a single successful result does not show that no other intersections exist. Examine the graph across the interval of interest and search for additional meeting points, using separate starting locations when the tool requires one.
Worked Example: Finding Two Intersections
Find the intersections of \(y=x^2\) and \(y=2x+3\).
Step 1: Create the difference.
Step 2: Locate the candidates. Factoring gives \(h(x)=(x-3)(x+1)\), so the candidate inputs are \(x=3\) and \(x=-1\). A calculator graph of the two functions, or a zero-solving tool applied to \(h\), should locate both candidates. Searching in regions around each input is a practical way to make sure the two displayed points are not mistaken for one another.
Step 3: Check the first candidate in both functions. At \(x=-1\), the first function gives \((-1)^2=1\). The second gives \(2(-1)+3=-2+3=1\). The values agree, so one intersection is \((-1,1)\).
Step 4: Check the second candidate in both functions. At \(x=3\), the first function gives \(3^2=9\). The second gives \(2(3)+3=6+3=9\). The values agree, so the other intersection is \((3,9)\).
Check: Substitution into each original function confirms both points: \(f(-1)=g(-1)=1\) and \(f(3)=g(3)=9\). The difference factors into exactly these two linear factors, so this equation has no other real zeros and these are all its intersections.
A Near Miss Is Not an Intersection
Displayed graph pixels have limited resolution, and a viewing window can make a small vertical gap hard to see. Likewise, a calculator may show two nearly equal output values after rounding. Such evidence can suggest a candidate, but intersection requires equality, not merely closeness. For a numerical candidate \(\hat{x}\), the difference \(h(\hat{x})=f(\hat{x})-g(\hat{x})\) measures how far apart the function values are at that input. A small difference is useful numerical evidence, but it is not automatically zero.
Worked Example: Graphs That Look as If They Meet
Compare \(y=x^2+0.01\) and \(y=x^2\). A graphing window with a broad vertical scale may make the graphs appear to overlap.
Step 1: Form the difference.
Step 2: Decide whether an intersection is possible. For an intersection, the difference would have to equal zero. Here \(h(x)=0.01\) for every real \(x\), so \(h(x)\) is never zero. There is no intersection.
Step 3: Check a particular input. At \(x=0\), the first graph has \(y=0^2+0.01=0.01\), while the second has \(y=0^2=0\). The vertical gap is \(0.01\), not zero. At \(x=2\), the values are \(2^2+0.01=4.01\) and \(2^2=4\), again separated by \(0.01\).
Check: The subtraction cancels \(x^2\) at every input and leaves \(0.01\), which cannot equal zero. The graphs may look close on screen, but both the algebraic difference and the substitutions show that they do not share a point.
Precision, Search, and Common Errors
A numerical solver works with finite precision, so its output is approximate. If it returns \(\hat{x}\), check the residual \(h(\hat{x})\), which is the difference between the two function values at that input. A small residual supports the candidate, but the number of displayed digits alone does not guarantee that all those digits are accurate. Re-evaluating both original functions at the same unrounded candidate is a useful check before rounding the final point.
One common mistake is to accept a graph’s visual appearance as proof. For the near-miss example, a student might reason, “The curves overlap on my screen, so they intersect.” That conclusion is not justified: the exact difference is \(0.01\), not zero. The correction is to solve or inspect \(f(x)-g(x)=0\), then verify the candidate in both original functions.
Another mistake is to calculate the \(y\)-coordinate using only one graph and never check the other. A reported point \((\hat{x},f(\hat{x}))\) lies on the first graph by construction, but it lies on the second only if \(f(\hat{x})\) and \(g(\hat{x})\) agree to the relevant precision. Also avoid rounding \(\hat{x}\) too early: changing the input can change both function values and make a valid candidate appear inconsistent. Keep the calculator’s more precise \(x\)-value for checking, and round only when stating the final coordinates.
Finally, do not conclude that no intersection exists just because a solver fails to find one. The tool may be searching in the wrong region, may have an unsuitable starting value, or may be affected by the window or input. A graph and a numerical solver help locate candidates; a conclusion about intersections must be supported by appropriate checks or mathematical reasoning. In an exam response, state the numerical method and give enough digits and substitution evidence for the reported coordinate to be understood and checked.
Check Your Understanding
Use the difference-function idea and the checks in this tutorial to answer each question.
- For graphs \(y=f(x)\) and \(y=g(x)\), what equation can you solve to search for their intersection \(x\)-coordinates?
- A calculator reports \(x\approx1.25\). What two function evaluations should you compare before reporting an intersection point?
- Suppose \(f(\hat{x})=4.002\) and \(g(\hat{x})=4.000\) on the calculator display. Why does this alone not establish an exact intersection?
- Why should you search more than one region of the graph when the functions may intersect multiple times?
- For \(f(x)=x^2+0.03\) and \(g(x)=x^2\), determine whether the graphs intersect and justify your answer using their difference.
