5.6 CONNECTING STANDARD & VERTEX FORM
Learning Goal: We will learn Partial Factoring to help us determine the axis of symmetry and eventually
the vertex of a parabola.
RECALL:
Standard Form: y = ax2 + bx + c
Factored Form: y = a (x - s)(x - t)
Vertex Form: y = a (x – h)2 + k
1.
Convert one form of the quadratic relation into another:
(i)
Factored to Vertex to Standard
y = (x + 3)(x – 2)
(ii)
Standard to Factored to Vertex
y =-3x2 + 6x + 45
Now try this…
2.
Determine the maximum value of the quadratic relation y = -3x2 + 12x + 29
THE PARTIAL FACTORING METHOD TO CONVERT THE QUADRATIC RELATION FROM
STANDARD FORM TO VERTEX FORM
If the quadratic relation cannot be completely factored:
1. Remove a partial (common) factor of x from the first 2 terms.
2. Find two points that are an equal distance from the axis of symmetry by
substituting y equal to the last value of the quadratic relation.
3. Determine the axis of symmetry by substituting the x values from the two
points that are an equal distance from the axis of symmetry into the formula
x  x2
x= 1
.
2
4. Substitute the x value for the axis of symmetry into the quadratic relation
and solve for y, to find the y coordinate of the vertex.
5. Substitute the coordinates of the vertex into the quadratic relation in vertex
form y = a(x – h)2 + k .
6. Substitute one of the points from step 2 to determine the value of “a”.
7. Substitute the value of “a” to complete the quadratic relation in vertex form.
Example:
1.
For the quadratic relation y = 2x2 + 8x + 5
i) Use partial factoring to two points that are an equal distance from the axis of
symmetry
ii) Find the coordinates of the vertex
iii) Express the relation in vertex form
iv) Sketch the graph
II)
APPLICATION OF QUADRATIC RELATIONS IN VERTEX FORM
Steps:
1) Attempt to remove a common factor equal to the numerical coefficient of the x2 term.
2) If no common factor for the entire polynomials is possible, use partial factoring to write
the quadratic relation in vertex form.
3) Use the value of h from the quadratic relation in vertex form to determine the value of x
at the quadratic relation’s maximum or minimum value.
4) Use the value of k from the quadratic relation in vertex form to determine the
maximum or minimum value of y.
Example:
2.
A stone is thrown off a cliff where its height, in h metres, after t seconds is
a)
What is the maximum height of the ball?
b)
When in the ball at its maximum height?
h = 27.3 + 4.8t – 2.4t2.