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.
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.
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
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
f(x) = a(x - h)² + k has a vertex at the point (h , k)
It is concave up when a>0
It is concave down when a<0
What is the value of the discriminant in the quadratic equation 2x² - 5x + 4 = 0 ?
discriminant =
discriminant = b² - 4ac
= (-5)² - 4x2x4
= 25 - 32
= -7
How many real roots does the function f have?
f(x) = 2x² + 12x + 18
number of roots =
discriminant = b² - 4ac
= (12)² - 4x2x18
= 0
Since discriminant = 0, there is one (repeated root)
We can see this if we write the function in the vertex from
f(x) = 2(x + 3)²
The vertex lies on the x axis at (-3 , 0)
The function below is given by f(x) = ax² + bx + c
Find a , b and c
a =
b =
c =
Since the x intercepts (or zeros) are at x = -3 and x = -1, we know that we can write the function in the form
f(x) = a(x + 3)(x + 1)
Let's expand the brackets
f(x) = a(x² + 4x + 3)
f(x) = ax² + 4ax + 3a
Since the y intercept is at y = 9, then 3a = 9
a = 3
f(x) = 3x² + 12x + 9
The function below is given by f(x) = ax² + bx + c
Find a , b and c
a =
b =
c =
Using the vertex (-3 , 4), we know
f(x) = a (x + 3)² + 4
Use the other point (0 , -14) to find a
-14 = a (0 + 3)² + 4
-14 = 9a + 4
-18 = 9a
a = -2
f(x) = -2 (x + 3)² + 4
f(x) = -2(x² + 6x + 9) + 4
f(x) = -2x² - 12x - 18 + 4
f(x) = -2x² - 12x - 14
The graph of the function f where f(x) = ax² + bx + c is given below
Drag the correct values into the table
negativezeropositive
a =
b =
c =
b² - 4ac =
b² - 4ac = 0 because there is one (repeated root).
c is positive. We can see the y intercept is positive.
a is positive. The graph is concave up.
b is positive. We can write the function in the vertex form f(x) = a(x - h)² . Since the vertex crosses the x axis where x<0, h must be negative. If you expand these brackets you will see that the x term is positive.
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
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
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