Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
Before you can solve an equation by computing, you can often solve it with your eyes! A curve carries a huge amount of information: every point on the curve of f has coordinates (x, f(x)). Knowing how to read a curve means answering questions about f with no computation at all.
Solving f(x) = k graphically means finding the x-coordinates of the points where the curve of f meets the horizontal line y = k. Each intersection point gives one solution: its x-coordinate. If there is no intersection point, the equation has no solution.
The same idea works for an equation like f(x) = g(x): the solutions are the x-coordinates of the intersection points of the two curves. Where the curves cross, the two functions take the same value.
For an inequality like f(x) < k, you no longer look for isolated points but for zones: the solutions are the x-coordinates of all the points where the curve lies strictly below the line y = k. The answer is then one or more intervals.
A pro tip: graphical reading gives approximate values or the number of solutions, while computation gives exact values. The two complement each other, and checking one against the other is an excellent habit.
The curve of f(x) = x² is drawn. Solve the equation f(x) = 4 graphically.
y = 4
I draw the horizontal line y = 4 on the same graph as the parabola.
2 points d'intersection
The line crosses the parabola at two points: the equation therefore has two solutions.
x = -2 et x = 2
I read the x-coordinates of these points: -2 and 2. Check by computing: (-2)² = 4 and 2² = 4.
Result: The solutions of f(x) = 4 are x = -2 and x = 2.
Still with f(x) = x², solve the inequality f(x) ≤ 1 graphically.
y = 1
I draw the horizontal line y = 1 and spot where the parabola is below or level with this line.
x² = 1 pour x = -1 ou x = 1
The parabola meets the line at the points with x-coordinates -1 and 1: these are the ends of the zone we want.
-1 ≤ x ≤ 1
Between these two x-values, the parabola sits under the line. The solution is the whole interval, endpoints included since the inequality is not strict.
Result: The solution set is the interval [-1, 1].