Solving Equations Graphically

Two graphs intersect exactly where their equations are simultaneously true, so the $x$-coordinates of intersections are the solutions. 0606 uses this both ways: drawing graphs to solve equations, and rearranging equations to reuse a graph you’ve already drawn.

The direct version

“On your diagram, draw a suitable line and hence solve $x^2 - 2x - 3 = x + 1$”: draw $y = x + 1$ over the existing parabola, mark the intersections, read the $x$-values. The marks: the correct added line (B), the solutions read to the expected accuracy (A). When the question says hence, the graphical route is the intended one, algebra may earn little or nothing.

The rearranging version, the real skill

You’ve drawn $y = x^2 - 2x - 3$. The question asks which line to add to solve $x^2 - 3x - 1 = 0$. Force the drawn function to appear:

$x^2 - 3x - 1 = 0$ $x^2 - 2x - 3 = x - 2$ (add $x - 2$‘s worth to both sides so the left side matches the drawn curve) Draw $y = x - 2$; the intersections solve the target equation.

Method: start from the target, add/subtract terms on both sides until one side is exactly the drawn function; whatever the other side became is the line to draw. The rearrangement line is the M mark, show it, don’t just announce the line.

Counting solutions

“State the number of solutions of $f(x) = k$” or “…of $f(x) = g(x)$”: count intersections on the sketch. Tangency counts once; this connects to the discriminant view, a line tangent to a curve meets it at one repeated solution. For modulus graphs the count-as-$k$-slides question is the standard form.

Accuracy expectations

Graphical solutions are read, not computed: quote them to the precision the grid supports (typically 1 d.p. on a drawn grid). Mark the intersections visibly, small crosses with dashed drop-lines to the axis show the examiner how you read the values, which protects the marks if your reading is slightly off.

Common mistakes

• Solving algebraically when the question demands the graphical route (“hence”, “using your graph”)
• The wrong line added because the rearrangement wasn’t done on paper
• $y$-coordinates quoted as solutions (solutions are $x$-values)
• Intersections counted but tangency double-counted
• No drop-lines/markings, unreadable working if the value is questioned

Full topic context: Equations, Inequalities & Graphs notes.