When two graphs cross, the intersection point is a solution to both equations at once — that’s how we use graphs to solve equations. The whole topic boils down to one move: plot, intersect, read off the coordinates on your GDC.
📘
What you need to know
The points of intersection of two graphs are the solutions to simultaneous equations
To solve f(x) = a: plot y = f(x) and the horizontal line y = a, then find intersections
To solve f(x) = g(x): plot both functions and find their intersections
The x-coordinates of the intersection points are the solutions
The number of intersection points = the number of solutions
Use the GDC’s “intersect” function for fast, accurate answers
Finding Intersection Points
Two graphs cross at a point only if the same (x, y) values satisfy both equations. So finding the intersection is the same as solving the simultaneous equations:
Two Lines, One Solution
The lines intersect at (2, 1) — and (x, y) = (2, 1) is the simultaneous solution of 2x − y = 3 and 3x + y = 7.
Plot both graphs on your GDC.
Use the intersect function (often under “Calc” or “Analyze graph”).
Read off the coordinates — the x-coordinate is the solution to the equation.
Adjust the window on your GDC if you don’t see all the intersection points — there may be more outside your initial view.
Using Graphs to Solve Equations
Two common situations:
Type 1
Solve f(x) = a
Plot y = f(x) and the horizontal liney = a. The x-coordinates of the intersections are the solutions.
Type 2
Solve f(x) = g(x)
Plot both curvesy = f(x) and y = g(x). The x-coordinates of the intersections are the solutions.
Counting the Number of Solutions
Some questions only ask for the number of solutions, not their values. Just count the points where the graphs cross:
0 intersections → no real solutions
1 intersection → 1 real solution
2 intersections → 2 real solutions
… and so on for 3, 4, etc.
Watch your GDC window: if you only see part of the graph, you might miss intersections at the edges. Zoom out to be sure.
Worked Examples
✎
Example 1 — Intersection of two lines
Find the point of intersection of the lines y = 2x + 1 and y = −x + 7.
Answer:
Plot both lines on the GDC and use the intersect tool.Set y-values equal: 2x + 1 = −x + 73x = 6 → x = 2y = 2(2) + 1 = 5(2, 5)
✎
Example 2 — Solve a quadratic graphically
Use a graph to solve x2 − 4x = 3. Give answers to 3 s.f.
Answer:
Plot y = x² − 4x and the horizontal line y = 3 on the GDC.Use the intersect tool at both crossing points.First intersection: x ≈ −0.646Second intersection: x ≈ 4.65x ≈ −0.646 or x ≈ 4.65The x-coordinates are your solutions — y-coordinates are not needed here.
✎
Example 3 — Number of solutions
Two functions are defined by f(x) = x3 − x and y = 2. Write down the number of real solutions to x3 − x = 2.
Answer:
Plot y = x³ − x and y = 2 on the GDC.Count intersection points.The cubic crosses y = 2 at one point (x ≈ 1.52).1 solutionNo need to find the actual value — just count crossings.
✎
Example 4 — Two curves intersecting
Two functions are defined by f(x) = x3 − x and g(x) = 4x. Find the coordinates where y = f(x) and y = g(x) intersect, to 3 s.f.
Answer:
Plot both curves on the GDC. Use the intersect tool at each crossing.Left intersection: (−1.60, −2.50)Right intersection: (1.60, 2.50)(−1.60, −2.50) and (1.60, 2.50)
✎
Example 5 — Solve from intersections
Using the same functions f(x) = x3 − x and g(x) = 4x, write down the solutions to x3 − x = 4x.
Answer:
Solutions of f(x) = g(x) are the x-coordinates of the intersections.From Example 4: intersections at x ≈ −1.60 and x ≈ 1.60x ≈ −1.60 or x ≈ 1.60Only the x-coordinates are needed for the equation solutions.
💡
Tips
Plot both graphs first, then use the GDC’s intersect tool. Don’t try to solve algebraically when a graph would be faster.
Check the window — if you don’t see enough of the graph, you might miss an intersection.
Quote answers as coordinates when asked for intersection points (x, y); quote just the x-values when asked for solutions.
Round to 3 s.f. unless the question says otherwise — and always show full GDC values in your working.
Sketch the graphs in your working, even on a calculator paper. Examiners give credit for visual reasoning.
⚠
Common mistakes
Reporting both x and y when only the equation solutions (x-values) were asked for.
Missing intersections at the edges of the GDC window. Always zoom out and check.
Forgetting to plot the second graph. To solve f(x) = 5, you need both y = f(x) AND y = 5 on the GDC.
Ignoring asymptotes when counting intersections. Two curves can come very close without actually crossing.
Rounding too early. Use the GDC’s full value, then round at the end. Premature rounding can give wrong final answers.
Confusing “number of solutions” with “find the solutions”. Read the question carefully — sometimes a count is enough.
Final word: Two graphs crossing = simultaneous equations solved. Plot, intersect, read off the coordinates — your GDC turns this whole topic into a one-step process.
Need help with Functions?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
