Answer: [tex]Vertex: y=\dfrac{1}{2}(x-2)^2-7[/tex]
[tex]Standard: y=\dfrac{1}{2}x^2-2x-5[/tex]
Transformations: vertical shrink by a factor of 1/2,
horizontal shift 2 units to the right,
vertical shift 7 units down.
Step-by-step explanation:
Vertex form: y = a(x - h)² + k
Standard form: y = ax² + bx + c
Given: Vertex (h, k) = (2, -7), the y-intercept (0, c) = (0, -5)
Input those values into the Vertex form to solve for the a-value
-5=a(0 - 2)² - 7
2 = a(- 2)²
2 = 4a
[tex]\dfrac{1}{2}=a[/tex]
a) Input a = 1/2 and (h, k) = (2, -7) into the Vertex form
[tex]\large\boxed{y=\dfrac{1}{2}(x-2)^2-7}[/tex]
b) You can plug in a = 1/2, c = -5, (x, y) = (2, -7) to solve for "b"
or
You can expand the Vertex form (which is what I am going to do):
[tex]y=\dfrac{1}{2}(x-2)^2-7\\\\\\y=\dfrac{1}{2}(x^2-4x+4)-7\\\\\\y=\dfrac{1}{2}x^2-2x+2-7\\\\\\\large\boxed{y=\dfrac{1}{2}x^2-2x-5}[/tex]
c) Use the Vertex form to describe the transformations as follows:
- a is the vertical stretch (if |a| > 1) or shrink (if |a| < 1)
- h is the horizontal shift (positive is to the right, negative is to the left)
- k is the vertical shift (positive is up, negative is down)
[tex]y=\dfrac{1}{2}(x-2)^2-7[/tex]
a = 1/2 --> vertical shrink by a factor of 1/2
h = 2 --> horizontal shift 2 units to the right
k = -7 --> vertical shift 7 units down
