1. Quadratic Functions
Vertex Form

2. Use the description to write the quadratic function in vertex form.
Example 1
Use the description to write the quadratic function in vertex form.
The parent function f(x) = x2 is vertically stretched by a factor of and then translated 2 units left and 5 units down to create g.
Step 1 Identify how each transformation affects the constant in vertex form.
Vertical stretch by : 4 3 a =
Translation 2 units left: h = –2
Translation 5 units down: k = –5

3. Step 2 Write the transformed function.
Example 1 continued
Step 2 Write the transformed function.
g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= (x – (–2))2 + (–5)
Substitute for a, –2 for h, and –5 for k.
= (x + 2)2 – 5
Simplify.
g(x) = (x + 2)2 – 5

4. Check Graph both functions on a graphing calculator
Check Graph both functions on a graphing calculator. Enter f as Y1, and g as Y2. The graph indicates the identified transformations. f g

5. Example 2
Use the description to write the quadratic function in vertex form.
The parent function f(x) = x2 is vertically compressed by a factor of and then translated units right and 4 units down to create g.
Step 1 Identify how each transformation affects the constant in vertex form.
Vertical compression by : a =
Translation 2 units right: h = 2
Translation 4 units down: k = –4

6. Step 2 Write the transformed function.
Example 2 Continued
Step 2 Write the transformed function.
g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= (x – 2)2 + (–4)
Substitute for a, 2 for h, and –4 for k.
= (x – 2)2 – 4
Simplify.
g(x) = (x – 2)2 – 4

7. Example 2 Continued
Check Graph both functions on a graphing calculator. Enter f as Y1, and g as Y2. The graph indicates the identified transformations. f g

8. Example 3
Use the description to write the quadratic function in vertex form.
The parent function f(x) = x2 is reflected across the x-axis and translated 5 units left and 1 unit up to create g.
Step 1 Identify how each transformation affects the constant in vertex form.
Reflected across the x-axis: a is negative
Translation 5 units left: h = –5
Translation 1 unit up: k = 1

9. Step 2 Write the transformed function.
Example 3 Continued
Step 2 Write the transformed function.
g(x) = a(x – h)2 + k
Vertex form of a quadratic function
= –(x –(–5)2 + (1)
Substitute –1 for a, –5 for h, and 1 for k.
= –(x +5)2 + 1
Simplify.
g(x) = –(x +5)2 + 1

10. Example 3 Continued
Check Graph both functions on a graphing calculator. Enter f as Y1, and g as Y2. The graph indicates the identified transformations. f g

11. Example 4: Scientific Application
On Earth, the distance d in meters that a dropped object falls in t seconds is approximated by d(t)= 4.9t2. On the moon, the corresponding function is dm(t)= 0.8t2. What kind of transformation describes this change from d(t)= 4.9t2, and what does the transformation mean?
Examine both functions in vertex form.
d(t)= 4.9(t – 0)2 + 0
dm(t)= 0.8(t – 0)2 + 0

12. Example 4 Continued
The value of a has decreased from 4.9 to 0.8. The decrease indicates a vertical compression.
Find the compression factor by comparing the new a-value to the old a-value.
a from d(t)
a from dm(t) =
0.8
4.9 
0.16
The function dm represents a vertical compression of d by a factor of approximately Because the value of each function approximates the time it takes an object to fall, an object dropped from the moon falls about 0.16 times as fast as an object dropped on Earth.

13. Example 5 Continued
Check Graph both functions on a graphing calculator. The graph of dm appears to be vertically compressed compared with the graph of d. 15 dm d

14. Example 5
The minimum braking distance d in feet for a vehicle on dry concrete is approximated by the function (v) = 0.045v2, where v is the vehicle’s speed in miles per hour.
The minimum braking distance dn in feet for a vehicle with new tires at optimal inflation is dn(v) = 0.039v2, where v is the vehicle’s speed in miles per hour. What kind of transformation describes this change from d(v) = 0.045v2, and what does this transformation mean?

15. Example 5 Continued
Examine both functions in vertex form.
d(v)= 0.045(t – 0)2 + 0
dn(t)= 0.039(t – 0)2 + 0
The value of a has decreased from to The decrease indicates a vertical compression.
Find the compression factor by comparing the new a-value to the old a-value. =
a from d(v)
a from dn(t)
0.039
0.045 13 15

16. Example 5 Continued
The function dn represents a vertical compression of d by a factor of The braking distance will be less with optimally inflated new tires than with tires having more wear.
Check Graph both functions on a graphing calculator. The graph of dn appears to be vertically compressed compared with the graph of d. 15 d dn