How do I sketch a graph of a polynomial function?

To sketch a polynomial function (a polynomial is a function involving powers of x, for example y = x3 + 2x2 - 13x + 10):

1. Start by differentiating the function to find its turning points (where the first differential is equal to zero). In this case, the first differential of the function is 3x2 + 4x -13. Solve for the roots of the differential (3x2 + 4x -13 = 0) by using the quadratic equation. The roots of the first differential are the points at which the graph has a maximum or minimum turning point (or, more unusually at A level, a point of inflection).

2.To check whether the turning point are maxima or minima, differentiate again for the second differential. Then put in the x-coodinates of each turning point. If the second differential is positive for that x-value, then the turning point is a minimum; if the second differential is negative then the point is a maximum.

3. Find the roots of the polynomial. Often, for a function with powers of x that are 3 or higher, an A-level question will give you a root; if not, try putting in values of x to the equation to find out if they are roots (a root is where y=0 on your graph). Once you have a root, use algebraic long division to factorise your polynomial. For example, with the polynomial y = x3 + 2x2 -13x +10, x=1 is a root, so you would divide (x3 + 2x2 -13x +10) by (x-1). This would give you (x2 + 3x - 10), which you can find the roots of using the quadratic equation. The graph crosses the x-axis at the roots - in this case, the graph crosses at x = -5, x = 1 and x = 2.

4. What does the graph do as x approaches infinity? Think about what happens as x gets very large and positive (going to the right on your graph). If the x term with the highest power is positive, then y will also be very large and positive. If the x term with the highest power is multiplied by a negative number, then y will be very large and negative. For example, for the funtion y = -x3 + 2x +1, as x becomes very large and positive, y becomes very large and negative, so the graph heads off downwards to the right after its last crossing point on the x-axis.

5. What happens as x approaches negative infinity? Again, think about what happens when x is very large and negative. This is a little more complicated because you also have to think about whether the highest power is odd or even. A negative value raised to a even power is positive, whilst a negative value raised to an odd power is negative. For example, when y = -x3 +2x + 1, as x becomes very large and negative, y becomes very large and positive. Here, x3 is negative when x is negative, but because x3 is multiplied by -1, y will be very large and positive when x approaches negative infinity.

6. When x is zero, what is y? It is always good to mark the y-intercept on your graph - just put x = 0 into the graph equation to find the value of the y-intercept.

Using all the information above, start your sketch. I'd usually start by marking the roots on the x-axis, then mark your turning points and join them up. Remember to only draw turning points you have found using the first differential - don't add any in by accident!

Rose A. A Level Physics tutor, A Level Maths tutor, GCSE Chemistry tutor

1 year ago

Answered by Rose, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Michael O.

Degree: MSci Theoretical Physics (Masters) - Birmingham University

Subjects offered: Maths, Physics+ 2 more

Further Mathematics
-Personal Statements-

“I'm a Theoretical Physics student at the University of Birmingham. In my eyes, our goal as Physicists is to understand everything and anything that the universe presents us with, and as such, is a subject which brings together so many...”

£20 /hr

Walter T.

Degree: Civil Engineering (Masters) - Bristol University

Subjects offered: Maths, Further Mathematics

Further Mathematics

“Hi! I'm Walter, a first year civil engineering student at the University of Bristol. I love maths and it forms a large part of my degree, so my love for it should come across easily in our sessions! I have experience as a tutor, having...”

MyTutor guarantee

£22 /hr

Oliver T.

Degree: Mathematics (Masters) - Edinburgh University

Subjects offered: Maths, Further Mathematics

Further Mathematics

“Hello! I'm currently a 2nd year Mathematics student at the University of Edinburgh with a sturdy passion for all things Mathematics. Not only do I love Maths, I love teaching Maths and helping people with problems. In particular, I en...”

About the author

£20 /hr

Rose A.

Degree: Physics (Masters) - Oxford, Keble College University

Subjects offered: Maths, Physics+ 1 more


“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Posts by Rose

How do I sketch a graph of a polynomial function?

Why does current split between branches of a parallel circuit, but voltage remains the same for each branch?

Why doesn't the concentration of products or reactants change when a reaction is at dynamic equilibrium?

Other A Level Maths questions

Show that 2sin(2x)-3cos(2x)-3sin(x)+3=sin(x)(4cos(x)+6sin(x)-3)

AS Maths ->Expresss x^2 + 3x + 2 in the form (x+p)^2 + q... where p and q are rational number

How do I multiply two matrices together?

How do we solve a second order, homogeneous, linear differential equation?

View A Level Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss