Quadratics

On this page we will look at quadratic expressions and how the different factorised forms link to the shape of the graph. We will also get an understanding about how the discriminant affects not only the number of roots of a quadratic equation, but also how we can use it as a tool to help us solve problems with intersections of graphs.


Key Concepts

On this page, you should learn to

  • Understand the quadratic function \(f(x)=ax^2+bx+c\) and its graph
  • Find and use the intercept form \(f(x)=a(x-p)(x-q)\)
  • Find and use the vertex form \(f(x)=a(x-h)^2+k\)
  • Solve quadratic equations
  • Solve quadratic inequalities
  • Use the discriminant \(\Delta =b^2-4ac\) to determine the nature of roots

Essentials

The following videos will help you understand all the concepts from this page

Quadratics and Graphs

In the following video, we look at quadratic expressions and their different factorised forms, how we might solve quadratic equations and most importantly how this all links to the graphs of the quadratic function.

Notes from the video

Quadratics and the Discriminant

In the following video, we are going to look at quadratic equations and the discriminant. For the general quadratic equation ax² + bx + c = 0, the discriminant is given by \(\triangle ={ b }^{ 2 }-4ac\)

We will see that if

  • b² – 4ac > 0 , there are 2 distinct real roots
  • b² – 4ac = 0 , there is 1 repeated real root
  • b² – 4ac < 0 , there are 0 real roots

Here you can play with the applet used in the video. Change the parameters of the quadratic function f(x) = ax² + bx + c and check the number of roots when the discriminant is positive, zero or negative.

Notes from the video

Discriminant Example

In the following video, we will look at a typical exam-style question involving quadratic functions. One of the challenges that we sometimes face with exam questions is that it is not always obvious what method we need to use to solve the problem. Often trying to visualise the situation with a graph helps. It certainly does in the question below:

Let f(x) = 2x² + kx + 3

Find the values of k for which f(x) > 0 , for all x

If you want to explore the graph in the question, you might find the following applet useful

Notes from the video

Discriminant and Intersecting Graphs

In the following video, we will look at a peculiar application of the discriminant.

If a quadratic function f meets the linear function g at a tangent, then the solution to the equation f = g has one (repeated) root. In other words, the f = g gives a quadratic equation and the discriminant = 0

Here is the question that we will consider to look at this case:


The line y = kx + 20 is a tangent to the curve f where f(x) = 12 -2x -2x²

Find the values of k.

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

The graph below is the function f(x)=a(x-h)²+k

The graph pass through the point (0,-9) and has vertex (2,3)

a) Write down the value of a, h and k

b) Find f(x) giving your answer in the form f(x) = Ax²+Bx+C

Hint

Full Solution

 

Question 2

Let f(x) = 2x²+12x +11

The function can also be expressed in the form f(x) = a(x-h)² +k

a) Find the equation of the axis of symmetry

b) Write down the value of h

c) Write down the value of k

Hint

Full Solution

 

Question 3

Let f(x) = x² + 2px + (3p+4)

Find the value of p so that f(x)=0 has two equal roots.

Hint

Full Solution

 

Question 4

Let f(x)=2x² + kx + 1 and g(x) = -x – 1.

The graphs of f and g intersect at two distinct points.

Find the possible values of k

Hint

Full Solution

 

Question 5

A quadratic function f can be written in the form f(x) = a(x - h)² + k . The graph of f has vertex (-1 , 4) and has y-intercept at (0 , -5).

a) Find the value of a , h and k.

b) The line y = mx + 1 is a tangent to the curve f. Find the value of m.

Hint

Full Solution

 

Just for Fun

If you understand the form y=a(x - p)(x - q) of a quadratic equation, then you should be able to play and win this game of angry birds. The angry bird starts at (0 , 0), the pig is at (10 , 0) and the bird must pass through (5,4).
Enter the equation in the space below and press play.

MY PROGRESS

How much of Quadratics have you understood?