Solve graphically

Intersections, how many solutions, above or below a level

In 3 of 20 recent exams
Paper 1 or 2
2 to 3 marks

How it is set in the exam

Paper 2 (calculator) more often (4 of the 7 parts set since 2021): the GDC's meeting points for 2–3 marks, usually with two models in a context. On Paper 1 (no calculator) it is a count or an inequality read off a sketch, for 1–2 marks at the end of a sketching question; a range of $k$ for a given number of solutions, read off a graph, is rare at SL. Typical (3 of 20 exam versions) as a part of its own, but the same GDC finish ends many other types: exponential models, segments, triangles, areas between curves.

Spot it in the exam

Exam phrases to spot (5)

“find all the times when the two heights are equal”
“hence write down the number of solutions of $f(x) = g(x)$ ”
“by sketching the graphs of $y = f(x)$ and $y = g(x)$ on the same axes, …”
“hence write down the values of $x$ for which $f(x) > 2$ ”
“find when the two objects meet”

In disguise (3)

"the distance between them is $0$ ": write the gap as the difference of the two positions, then find its zeros
two sinusoidal models equal to each other: both are Topic 3 models, but one equation between two models is solved here
a rational function and a parabola sketched together, and their meeting points counted (Rational functions works that chain)

Variants (4)

a count from a sketch ()
an inequality against a constant, read off a sketch ()
every meeting point on a domain, from the GDC: the complete set, in radians ()
moving objects: the gap between their positions first, then its zeros ()

Looks like it, but isn't (9)

an analytic method exists: Quadratic in disguise, Two graphs, or the exponential and log equations of Topic 1 (Solve an exponential equation)
a quadratic above or below a level or a line, solved exactly: Quadratic inequalities: when is it above or below?
$f(x) = k$ for an unknown $k$ (how many solutions there are for each $k$ ), with $k$ only in the constant and a formula for $f$ : Find the vertex for a parabola, Rational functions: asymptotes, intercepts, sketch, range (the range) for $\frac{ax + b}{cx + d}$ , Stationary points otherwise. From a given graph with no formula, it stays here (Example 3.1).
$f(x) = f^{-1}(x)$ : The graphs of f and f⁻¹
one trig ratio equal to a number: Solve a trig equation; a sinusoid against a constant level: Above or below a level
an exponential model reaching a value: Exponential and log models
where an exponential or log function is positive or negative (compared with $0$ ), from $e^{\ldots} > 0$ or the sign of $\ln$ : Exponential and log graphs
a GDC solve that finishes a Topic 3 set-up ( $\theta$ and $\sin\theta$ after a segment; a system from a triangle's area and the cosine rule): the part belongs to Chords, segments and regions or Area of a triangle; this type supplies only the last step
positions that must first be found from velocities (integrate, then set them equal): Displacement, position and distance travelled

Worked example

A count and an inequality from a given graph

Paper 1 (no calculator)

Question

The graph of $y = f(x)$ , for $-3 \le x \le 6$ , is shown. It starts at $(-3, -1)$ , rises through $(-2, 0)$ to a local maximum at $(0, 4)$ , falls through $(2, 0)$ to a local minimum at $(4, -2)$ , then rises through $(5, 0)$ to end at $(6, 1)$ . It has no other turning points.
(a) Write down the number of solutions of $f(x) = 3$ . [1]
(b) Find the values of $k$ for which the equation $f(x) = k$ has exactly three solutions. [2]
(c) Write down the values of $x$ for which $f(x) \ge 0$ . [2]

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Which method?

The question	Use
: how many solutions, or where $f$ is above or below a constant	a sketch: place the features that decide it (turning points, ends, asymptotes), then count the meeting points or read the crossings
: solve $f(x) = g(x)$	the GDC: graph both on the domain and find every intersection (zeros and intersections)
: two moving objects meet	the gap $d(t) = s_1(t) - s_2(t)$ , then its zeros (the same as intersecting the two positions)
a system of two equations	substitute one into the other to get one equation in one variable, or give both the same subject (both $y = \ldots$ , or both $y^2 = \ldots$ ) and intersect the graphs: two graphs with different subjects lose the method mark

Steps

Write one equation,
$f(x) = g(x)$ , in one variable, with its domain.
Graph both sides on the domain:
a sketch on , the GDC on , in radians for any trig (check the settings).
Read every meeting point,
or count them, or read the interval where $f$ is above or below the constant. On the GDC, find the intersections one at a time until the whole domain is covered.
Keep only what the domain or the context allows,
say why the others go, and answer in the terms asked: a time, a position, a count, an interval.

Common mistakes

An incomplete or padded set
"find all" is marked as a set. One missing solution, or one extra from outside the domain, loses the final marks. Scan the whole domain, one intersection at a time.
Degree mode
in a trig model: no solutions, or the wrong ones.
Solutions outside the domain or the model kept
negative times, or times after the model stops. Reject them, and say why.

More mistakes (5)

A count from a sketch that is wrong where it matters
the count depends on whether the line passes above or below a turning point, or through an end, so place those carefully. Follow-through does not rescue a count: both graphs on the same axes with the meeting points marked keep the method mark, but a count read off a wrong earlier sketch loses the answer mark.
Half an equation graphed
making $y$ the subject of an equation with $y^2$ in it gives two functions, $y = \pm\sqrt{\ldots}$ . Graph both, or substitute instead.
The ends of a closed domain forgotten
a horizontal line through an included end meets the graph there (Example 3.1(b)).
One interval written across a gap
$-2 \le x \le 6$ in place of $-2 \le x \le 2$ or $5 \le x \le 6$ .
Answering a different question
the time, when a position or a height was asked for.

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