Finding Rates of Change –

Part 2

Slideshow 30, MathematicsMr. Richard Sasaki, Room 307

Objectives• Be able to find the range and domain

for quadratic graphs• Be able to draw quadratic graphs in

vertex form based on their rate of change

• Use this to name quadratic graphs

Range and DomainWe learned about these two words last year. Do you remember them?Range - A data set designed for Domain - A data set

designed for

Domain

Range

Basically, the domain refers to part of the – axis and the range refers to part of the – axis.

Range and DomainThe process for quadratic graphs is exactly the same.ExampleLook at the graph below.

Consider a range of and write down its corresponding domain.Domain:−4≤ 𝑥≤−2

How about the entire range for a domain ?−4≤ 𝑓 (𝑥 )≤∞⇒ 𝑓 (𝑥)≥−4

Answers - Easy1. , 2. , (all real numbers)3. , 4. , and

Answers – Medium1. ,

2. ,

Answers – Hard

1. ,

2. ,

Drawing Graphs with Rate of ChangeFor the last of our 3 methods of drawing quadratics, we will learn the quickest method! For this we consider gradient triangles.

11315

11315

The rate of change is identical for any graph where .

Rate of ChangeWhat would the rate of change be when ?

1216

110

It’s always double the gradient!In fact the rate of change from the vertex always goes for any quadratic.

22

6Be careful when . A triangle does not have the same effect as a triangle.

When , why ?

1𝑎=12

13𝑎=

32

Answers - EasyReally they are following the relationship (square numbers ) for bases of respectively. If each has a base of 1, we subtract previous heights.

Answers - Hard

The Effects of , and The influence of , and when the line is in vertex form is much more obvious than , and in regular form. Look here (Internet necessary.)To summarise…

.affects the rate of change (steepness) .affects the position (anticlockwise) .affects the position (vertically)

causes the entire curve to move in a way that forms the shape of a negative curve., and have a much simpler effect on the line. Look here (Internet necessary.)

Naming Quadratic GraphsWriting the name of a graph in vertex form is useful as the vertex is easy to spot.ExampleName the graph below. Write it in vertex form.

(4 ,−5)

𝑦=𝑎 (𝑥−h )2+𝑘𝑦=𝑎 (𝑥−4 )2−5

12

𝑦=2 (𝑥−4 )2−5

(when the base is 1)

That’s all! Just expand to write it in its regular form!

Answers – Part 1(𝑥−3 )2+1 2 (𝑥−2 )2

2 (𝑥+4 )2−3(𝑥−1 )2

2

𝑦=2 (𝑥−1 )2+2𝑦=

−𝑥23

Answers – Part 2 (Easy)2 𝑥2−12𝑥+19𝑥2−4 𝑥+4

2 𝑥2+16 𝑥+313 𝑥2+30𝑥+75

𝑦=𝑥2−6𝑥+12 , 𝑥=3𝑦=

𝑥22−𝑥+

32

Answers – Part 2 (Hard)−𝑥2+2𝑥−13 𝑥2+42 𝑥+141

− 𝑥2

3+6 𝑥−33

𝑥26−𝑥− 7

2

𝑦=𝑥2−10 𝑥+32𝑦=

−𝑥24− 3𝑥2

+234