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Module 19.1 Understanding Quadratic Functions P. 889 What is the effect of the constant a on the graph of ?
Module 19.1
Understanding Quadratic Functions
P. 889
What is the effect of the constant a on the graph of 𝒂𝒙𝟐?
Do you remember these two fromlinear functions? (Module 6.4)
P. 889
What does Quadratic mean?The degree (power) is 2, as in 𝒙𝟐.
What is a Quadratic Function?A function that can be represented in the form 𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄,where:
𝒂, 𝒃, and 𝒄 are constants𝒂 cannot equal zero.
This is called STANDARD FORM.
a b c f(x)
1 1 1 𝑥2 + 𝑥 + 1
1 2 3 𝑥2 + 2𝑥 + 3
3 4 5 3𝑥2 + 4𝑥 + 5
2 3 0 2𝑥2 + 3𝑥
1 0 5 𝑥2 + 5
3 0 0 3𝑥2
1 0 0 𝑥2
Examples:
Parent Quadratic Function (the most basic)
𝒇 𝒙 = 𝒙𝟐 𝒘𝒉𝒆𝒓𝒆 𝒃 𝒂𝒏𝒅 𝒄 = 𝟎
𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
P. 889
Why can the y value never be negative?
P. 889
Vertex - where the curve turns. In this case – the origin - (0,0).aka Minimum
Axis of Symmetry
Parabola
A vertical line that passes through the vertex and divides the parabola into two symmetrical halves, mirror images of each other.
In this case – the Y axis.𝒇 𝒙 = 𝒙𝟐
𝒇 𝒙 = 𝒙𝟐
x f(x)
-5 25
-4 16
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
4 16
5 25
𝑓 𝑥 = 𝑎𝑥2 𝑤ℎ𝑒𝑟𝑒 𝑎 = 1 𝒇
Domain: All real numbersRange: All y≥0
Because both side of the parabola are equal, you only have to calculate one set of points.
𝒈 𝒙 = 𝟐𝒙𝟐
x g(x)
-3 18
-2 8
-1 2
0 0
1 2
2 8
3 18
𝑔 𝑥 = 𝑎𝑥2 𝑤ℎ𝑒𝑟𝑒 𝑎 = 2
Domain: All real numbersRange: All y≥0
𝒈 𝒙 = 𝟑𝒙𝟐
x g(x)
-2 12
-1 3
0 0
1 3
2 12
𝑔 𝑥 = 𝑎𝑥2 𝑤ℎ𝑒𝑟𝑒 𝑎 = 3
Domain: All real numbersRange: All y≥0
𝒈 𝒙 = 𝟒𝒙𝟐
x g(x)
-2 16
-1 4
0 0
1 4
2 16
𝑔 𝑥 = 𝑎𝑥2 𝑤ℎ𝑒𝑟𝑒 𝑎 = 4
Domain: All real numbersRange: All y≥0
As a increases from 1, the graph becomes more narrow. This is called a Vertical Stretch. Rubber band.
𝒇
𝒈 𝒙 = 𝒂𝒙𝟐
a = 2𝒈 𝒙 = 𝒂𝒙𝟐
a = 3
𝒈 𝒙 = 𝒂𝒙𝟐
a = 4
𝒇 𝒙 = 𝒂𝒙𝟐
a = 1Parent Function
What happens when a is between 0 and 1?
𝒈 𝒙 = ½𝒙𝟐
x g(x)
-6 18
-4 8
-2 2
0 0
2 2
4 8
6 18
𝑔 𝑥 = 𝑎𝑥2 𝑤ℎ𝑒𝑟𝑒 𝑎 = ½
Domain: All real numbersRange: All y≥0
𝑔 𝑥 = 𝑎𝑥2 𝑤ℎ𝑒𝑟𝑒 𝑎 = ⅓
𝒈 𝒙 = ⅓𝒙𝟐
x g(x)
-9 27
-6 12
-3 3
0 0
3 3
6 12
9 27
Domain: All real numbersRange: All y≥0
𝑔 𝑥 = 𝑎𝑥2 𝑤ℎ𝑒𝑟𝑒 𝑎 = ¼
𝒈 𝒙 = ¼𝒙𝟐
x g(x)
-10 25
-8 16
-6 9
-4 4
-2 1
0 0
2 1
4 4
6 9
8 16
10 25
Domain: All real numbersRange: All y≥0
As a decreases from 1 to 0, the graph becomes wider. This is called a Vertical Compression..
𝒇 𝒙 = 𝒂𝒙𝟐
a = 1Parent Function
𝒇
𝒈 𝒙 = 𝒂𝒙𝟐
a = ½𝒈 𝒙 = 𝒂𝒙𝟐
a = ⅓𝒈 𝒙 = 𝒂𝒙𝟐
a = ¼
The domain of a quadratic function is all real numbers.
When a > 0:▪ The graph of 𝒈 𝒙 = 𝒂𝒙𝟐 opens upward▪ The function has a minimum value that occurs at the vertex▪ The Range is all y≥0
Summary:The smaller a is, the wider the graph is. The larger a is, the thinner the graph is.
Vertical StretchVertical Compression
P. 891
𝒈 𝒙 = 𝟑𝒙𝟐
x g(x)
P. 891
4. 𝒈 𝒙 = 𝟑𝒙𝟐
P. 891
𝒈 𝒙 = ⅓𝒙𝟐
x g(x)
What happens when a is less than 0?
a = −½a = −⅓a = −¼
When a < 0:▪ The graph of 𝒈 𝒙 = 𝒂𝒙𝟐 opens downward▪ The function has a maximum value that occurs at the vertex▪ The Range is all y≤0
a = −𝟐 a = −3 a = −4a = −𝟏
Again:The smaller a is, the wider the graph is. The larger a is, the thinner the graph is.
Vertical StretchVertical Compression
a = −𝟐 a = −3 a = −4a = −𝟏
a > 0
a = −½a = −⅓a = −¼
𝒇 𝒙 = 𝒂𝒙𝟐
a = 𝟏a = 𝟐 a = 𝟑 a = 𝟒
a < 0
𝒈 𝒙 = 𝒂𝒙𝟐
a = ¼ a = ⅓ a = ½
minimum
maximum
P. 908
Q & A
How can you immediately tell from the graph that the value of a in the quadratic function is positive? The graph opens upward.
How can you immediately tell from the graph that the value of a in the quadratic function is negative? The graph opens downward.
𝒈 𝒙 = 𝒂𝒙𝟐
𝒈 𝒙 = −𝒂𝒙𝟐
P. 893
P. 893
P. 893P. 894
P. 894
P. 894
P. 895
