Day 4: Graph y = a(x – h)2 + k Part 1
1. How can you tell if a parabola opens up or down?
2. The _______________ is the highest or lowest point on the graph of a parabola.
3. When a quadratic equation is in the form y= a(x-h)2 + k, we call this vertex form. This is
because it is easy to determine the coordinates of the vertex of the parabola. The vertex is
____________ when the equation is in vertex form y= a(x-h)2 + k.
4. The “centre line” of each parabola is called its ____________ of _________________.
5. The ____________________ always lies on the axis of symmetry.
6. The points where the parabola crosses the x-axis are called the x-intercepts or _________
of the relation.
Summary of Graphing Equations in Vertex Form
y= a(x-h)2 + k
Operation
Resulting Equation
Multiply by a
y = ax2
Replace x by (x-h)
Add k
y= (x-h)2
y= x2 + k
What happens
If a > 0, parabola opens up.
If a< 0, parabola opens down.
This transformation is called a vertical reflection.
If a > 1 or a< -1, parabola gets thinner
This transformation is called a vertical stretch.
If -1 < a < 1, parabola gets wider.
This transformation is called a vertical compression.
If h > 0 (“-“ in the brackets),
parabola moves h units to the right.
This transformation is called a horizontal translation.
If h < 0 ( “+” in the brackets),
parabola translates h units to the left.
This transformation is called a horizontal translation.
If k > 0, parabola moves k units up.
This transformation is called a vertical translation.
If h < 0, parabola moves k units down.
This transformation is called a vertical translation.
Example 1:
Complete the chart.
Function
Direction
of
Opening
Coordinates
of Vertex
Narrow or
Wide?
Equation of
Axis of
Symmetry
Maximum
or minimum
value
y= 3(x-1)2 + 2
y= -(x-4)2 + 1
1
(x+2)2 - 3
4
y= 2x2 + 4
y= -
y= 4(x-5)2
Example 2:
Graph each quadratic relation and fill in the chart.
a) y = -x2 + 3
Property
vertex
Axis of symmetry
Stretch or compression
factor relative to
y = x2
Direction of opening
Values x may take
Values y may take
Note: The order of transformations follows the order of operations. Stretches,
compressions and reflections are done before translations!
1
b) 𝑦 = 2 (𝑥 + 4)2 − 3
Property
vertex
Axis of symmetry
Stretch or compression
factor relative to
y = x2
Direction of opening
Values x may take
Values y may take
Example 3:
Determine the equation of the given parabola.
Example 4:
Write an equation for the parabola with vertex (-4, 5), opening upward,
2
with a vertical compression factor of 5.
Challenge:
Page 188 #19, 20