Quadratic Functions

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.

Use the description to write the quadratic function in vertex form.

Example 1

Step 1 Identify how each transformation affects the constant in vertex form.

Translation 2 units left: h = –2

Translation 5 units down: k = –5

4

3

Vertical stretch by :4

3a4

3

Example 1 continued

Step 2 Write the transformed function.

g(x) = a(x – h)2 + k Vertex form of a quadratic function

Simplify.

= (x – (–2))2 + (–5) 4

3Substitute for a, –2 for h, and –5 for k.

4

3

= (x + 2)2 – 5 4

3

g(x) = (x + 2)2 – 5 4

3

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

f

g

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 2 units right and 4 units down to create g.

Step 1 Identify how each transformation affects the constant in vertex form.

Translation 2 units right: h = 2

Translation 4 units down: k = –4

Vertical compression by : a =

Step 2 Write the transformed function.

g(x) = a(x – h)2 + k Vertex form of a quadratic function

Simplify.

= (x – 2)2 + (–4)

= (x – 2)2 – 4

Substitute for a, 2 for h, and –4 for k.

Example 2 Continued

g(x) = (x – 2)2 – 4

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

f

g

Example 2 Continued

The parent function f(x) = x2 is reflected across the x-axis and translated 5 units left and 1 unit up to create g.

Example 3

Use the description to write the quadratic function in vertex form.

Step 1 Identify how each transformation affects the constant in vertex form.

Translation 5 units left: h = –5

Translation 1 unit up: k = 1

Reflected across the x-axis: a is negative

Step 2 Write the transformed function.

g(x) = a(x – h)2 + k Vertex form of a quadratic function

Simplify.

= –(x –(–5)2 + (1)

= –(x +5)2 + 1

Substitute –1 for a, –5 for h, and 1 for k.

Example 3 Continued

g(x) = –(x +5)2 + 1

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

Example 3 Continued

f

g

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

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 0.16. 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.

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

15

150

0

dm

d

Example 5 Continued

Example 5

The minimum braking distance dn in feet for avehicle 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?

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.

Examine both functions in vertex form.

d(v)= 0.045(t – 0)2 + 0 dn(t)= 0.039(t – 0)2 + 0

Example 5 Continued

The value of a has decreased from 0.045 to 0.039. 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) 0.0390.045

=1315

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

150

0

d

dn

Example 5 Continued