Sketch the curve y = x^2 - 6x + 5, identifying roots and minima/maxima.

Remeber the formula: (a - b)2 = a2 - 2ab + b2. Notice that y = x2 - 23x + 5, so we want to write this using (x - 3)2 = x2 - 23x* + 9. Taking 4 from both sides gives:  (x - 3)2 - 4 = x2 - 6x + 5 = y.

We need some simple facts about graphs: (1) y = x2 is a parabola (U shaped); (2) if we replace x wih x - 3 we move the graph to the right by 3; (3) if we add -4 to y, the graph moves down by 4.

To find minima: notice that (x - 3)2 is always positive or 0, so (x - 3)2 + -4 >= -4. If x is not 3, then (x - 3)2 > 0, so y > -4; but if x = 3, we have y = -4, so -4 is the smallest value of y (i.e. a minimum) at (3, -4).

To find roots, we can solve the quadratic y = 0:

(x - 3)2 - 4 = 0  <=>  (x - 3)2 = 4  <=>   x - 3 = 2  or  x - 3 = -2  <=>  x = 5  or  x = 1.

(Rememer that x2 = a2 has two solution: x = a and x = -a.) With this it should be easy to sketch the curve!

Answered by Tutor69809 D. Maths tutor

4264 Views

See similar Maths GCSE tutors

Related Maths GCSE answers

All answers ▸

Work out the gradient of the tangent to the curve (y=x^2-x-2) at the point where x=2


Solve the simultaneous equations x + y = 3 and x^2 + y^2 = 5


ABC, DEF and PQRS are parallel lines. BEQ is a straight line. Angle ABE = 60° Angle QER = 80° Work out the size of the angle marked x. Give reasons for each stage of your working.


Make y the subject of the formula 3y-p=h(2+y)


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo
Cookie Preferences