OBJECTIVE To plot two functions (either a linear and quadratic or two quadratics) and to fi nd their points of intersection both graphically and geometrically
Y 6 EXAMPLE 13
A road with equation y = f (x) = x 2 – 5x + 7 is passing through a town. The road that bypasses the town is a motorway with equation y = g (x) = x + 2. Find the points where the road intersects the motorway. If all of the distances are in kilometres (km) and the speed limit is 120 km/h, fi nd the least time a car can travel between these points. Give the time in minutes correct to one decimal place.
Solution y = g (x) = x + 2
y = f (x) = x 2 – 5x + 7 x + 2 = x 2 – 5x + 7 x 2 – 6x + 5 = 0 (x − 1)(x − 5) = 0 x = 1, 5
Substituting into g: y = 3, 7 ∴ A(1, 3) B(5, 7) |AB| = √
_______ 16 + 16 = √
v = s _
t ⇒ t = s __
t = √
___ 30 × 60 minutes
__ 2
= 2 √ 262
__ 2 minutes
= 2⋅8 minutes v = √
____ 120 = √
___ 32
___ 32 km
___ 30 hours
__ 2
A g (x) f (x) B Algebraically: Equate the y co-ordinates and solve the resulting equation.
Graphically: Plot the functions on the same graph paper and read off their points of intersection.
