What to Learn 
Help to Remember What to Learn

y = a(x  h)^{2} + k
 If a < 0, the quadratic will open downwards.
 If a > 0, the quadratic will open upwards.
 Vertical Stretch:
 Increasing the value of a will vertically
expand the graph (ie, make the arms move closer together).
 Decreasing the value of a will
vertically compress the graph (ie, make the arms spread out wider).
 When you move a to a negative value you will
see that there is a reflection about the xaxis. Looking at the
change in a relative to the xaxis will help
you see the expansion/contraction better.
 (h, k) will always be the coordinates
of the vertex (the minimum/maximum value) of the quadratic function.
 Domain: xR.
 Range:
 if a > 0, y > k
 if a < 0, y < k
 Note: When the form changes to a(x + h)^{2} +
k, then (h, k) is the
vertex, BUT, h is a negative number.
 Note: When the form changes to a(x + h)^{2}  k,
then (h, k) is still the vertex
BUT k will have a negative value.

Quadratic Standard (Completed Square) Form Interactive
Activity
Move the sliders for a, h, and k.
Slider b will not be studied in Math 20 Pure.
 Study the effect of changing a from positive to
negative values.
 Study the effect changing h and k has
on the vertex (the minimum/maximum value) position.
 Watch the operation for h in the (x 
h) section of the top left corner of the data
output.
 Find the xintercept(s) on the graph. These will be the solutions
of the form: y = a(x  h)^{2} + k
 Find the yintercept on the graph. Notice that the a, b and k parameters
do not help you determine the location of the yintercept.
Linked
Source  Ron Blond

Question 1:
Find the equation of a quadratic function given the vertex, (2, 4)
and the yintercept of 4.

Sample Solution
y = a(x  h)^{2} + k
The question provides the vertex, (2, 4). Thus (h, k) = (2, 4).
y = a(x  2)^{2}  4
The yintercept, (0, 4), is on the quadratic. Thus
(x, y) = (0, 4). Substitute these values into the equation:
y = a(x  2)^{2}  4
4 = a [(0)  2]^{2}  4
a (2)^{2} = 4+4
4a = 8
a = 2
Therefore, the equation of the quadratic is y = 2(x  2)^{2} 
4 Note: The steps are similar if given the vertex and another point
on the quadratic.

Question 2:
Write an equation for a parabola with vertex (3, 4) and passing through
(2, 7). 
Sample Solution
y = a(x  h)^{2} + k
The question provides the vertex, (3, 4). Thus (h, k) = (3, 4).
y = a(x  3)^{2}  4
(2, 7) is on the quadratic. Thus (x, y) = (2, 7).
Substitute these values into the equation:
y = a(x  3)^{2}  4
7 = a [(2)  3]^{2}  4
7 = a (1)^{2}  4
a (1)^{2} = 7 + 4
a = 3
Therefore, the equation of the quadratic is y = 3(x  3)^{2} 
4
Note: The steps are similar if given the vertex and another point
on the quadratic. 
Enrichment: Quadratic
General to Standard Form Proof
 Domain: xR.
 Range:
 if a > 0, y > k
 if a < 0, y < k

