Quadratic Functions & Graphs
What are the key features of quadratic graphs?
- A quadratic graph can be written in the form where
- The shape of the graph is known as a parabola
- The value of a affects the shape of the curve
- If a is positive the shape is 'u-shaped' ∪
- If a is negative the shape is 'upside down u-shaped' ∩
- The y-intercept is at the point (0, c)
- The roots are the solutions to
- These are also known as the x-intercepts or zeroes
- They can be found by
- Factorising
- Quadratic formula
- Completing the square
- Your calculator may also be able to find these for you
- There can be 0, 1 or 2 x-intercepts
- This is determined by the value of the discriminant
- There is an axis of symmetry at
- If there are two x-intercepts then the axis of symmetry goes through the midpoint between them
- The vertex lies on the axis of symmetry
- It can be found by completing the square
- The x-coordinate is
- The y-coordinate can be found by calculating y when
- If a is positive then the vertex is the minimum point
- If a is negative then the vertex is the maximum point
What are the equations of a quadratic function?
-
- This is the general form
- It clearly shows the y-intercept (0, c)
- You can find the axis of symmetry by
-
- This is the factorised form
- It clearly shows the roots (p, 0) & (q, 0)
- You can find the axis of symmetry by
-
- This is the vertex form (or completed square form)
- It clearly shows the vertex (h, k)
- The axis of symmetry is therefore
- It clearly shows how the function can be transformed from the graph of
- Vertical stretch by scale factor a
- Translation by vector
How do I find an equation of a quadratic?
- If you have the roots x = p and x = q...
- Write in factorised form
- You will need a third point to find the value of a
- If you have the vertex (h, k) then...
- Write in vertex form
- You will need a second point to find the value of a
- If you have three random points (x1, y1), (x2, y2) & (x3, y3) then...
- Write in the general form
- Substitute the three points into the equation
- Form and solve a system of three linear equations to find the values of a, b & c
Exam Tip
- Your calculator may be able to find the roots and turning point of a quadratic function
- Even on a 'show that' question this can be used to check your answers
Worked example
The diagram below shows the graph of , where is a quadratic function.
The vertex and the intercept with the -axis have been labelled.
Find an expression for .
Method 1
Since we know the vertex (turning point), it will be easiest to start with the completed square version of the equation
This is , where the vertex is at
We also know the curve goes through (0, 6)
Put those coordinates into the equation and solve for
Substitute into the expression for
Expand the brackets and rearrange into the form required
Method 2
It is also possible to start with the form
Because the y-intercept is (0, 6) we know that
Goes through means
It also goes through (-1, 8)
Substitute those coordinates into the equation of
Goes through means
We need one more piece of information
You may remember that the turning point lies on the line
If not, then use the fact that the x-coordinate of the turning point satisfies
Turning point at means
Differentiate to find
Then solve to find another equation with and
We now have two simultaneous equations that we can solve to find and
Add [1] and [2] together to eliminate
Substitute into [2] and solve to find
Write final answer in form requested