Skip to content Skip to navigation Skip to collection information

OpenStax-CNX

You are here: Home » Content » Advanced Algebra II: Activities and Homework » Graphing Quadratic Functions II

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them in […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Busbee's Math Materials display tagshide tags

    This collection is included inLens: Busbee's Math Materials Lens
    By: Kenneth Leroy Busbee

    Click the "Busbee's Math Materials" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Graphing Quadratic Functions II

Module by: Kenny M. Felder. E-mail the author

Summary: Further problems on graphing quadratic functions.

Yesterday we played a bunch with quadratic functions, by seeing how they took the equation y=x2y=x2 size 12{y=x rSup { size 8{2} } } {} and permuted it. Today we’re going to start by making some generalizations about all that.

Exercise 1

y = x 2 y = x 2 size 12{y=x rSup { size 8{2} } } {}

  • a. Where is the vertex?
  • b. Which way does it open (up, down, left, or right?)
  • c. Draw a quick sketch of the graph.

Exercise 2

y = 2 ( x 5 ) 2 + 7 y = 2 ( x 5 ) 2 + 7 size 12{y=2 \( x - 5 \) rSup { size 8{2} } +7} {}

  • a. Where is the vertex?
  • b. Which way does it open (up, down, left, or right?)
  • c. Draw a quick sketch of the graph.

Exercise 3

y = ( x + 3 ) 2 8 y = ( x + 3 ) 2 8 size 12{y= \( x+3 \) rSup { size 8{2} } - 8} {}

  • a. Where is the vertex?
  • b. Which way does it open (up, down, left, or right?)
  • c. Draw a quick sketch of the graph.

Exercise 4

y = ( x 6 ) 2 y = ( x 6 ) 2 size 12{y= - \( x - 6 \) rSup { size 8{2} } } {}

  • a. Where is the vertex?
  • b. Which way does it open (up, down, left, or right?)
  • c. Draw a quick sketch of the graph.

Exercise 5

y = x 2 + 10 y = x 2 + 10 size 12{y= - x rSup { size 8{2} } +"10"} {}

  • a. Where is the vertex?
  • b. Which way does it open (up, down, left, or right?)
  • c. Draw a quick sketch of the graph.

Exercise 6

Write a set of rules for looking at any quadratic function in the form y=a(xh)2+ky=a(xh)2+k size 12{y=a \( x - h \) rSup { size 8{2} } +k} {} and telling where the vertex is, and which way it opens.

Exercise 7

Now, all of those (as you probably noticed) were vertical parabolas. Now we’re going to do the same thing for their cousins, the horizontal parabolas. Write a set of rules for looking at any quadratic function in the form x=a(yk)2+hx=a(yk)2+h size 12{x=a \( y - k \) rSup { size 8{2} } +h} {} and telling where the vertex is, and which way it opens.

After you complete #7, stop and let me check your rules before you go on any further.

OK, so far, so good! But you may have noticed a problem already, which is that most quadratic functions that we’ve dealt with in the past did not look like y=a(xk)2+hy=a(xk)2+h size 12{y=a \( x - k \) rSup { size 8{2} } +h} {}. They looked more like…well, you know, x22x8x22x8 size 12{x rSup { size 8{2} } - 2x - 8} {} or something like that. How do we graph that?

Answer: we put it into the forms we now know how to graph.

OK, but how do we do that?

Answer: Completing the square! The process is almost—but not entirely—like the one we used before to solve equations. Allow me to demonstrate. Pay careful attention to the ways in which is is like, and (more importantly) is not like, the completing the square we did before!

Table 1
Step Example
The function itself x 2 6x 8 x 2 6x 8 size 12{x rSup { size 8{2} } - 6x - 8} {}
We used to start by putting the number (–8 in this case) on the other side. In this case, we don’t have another side. But I still want to set that –8 apart. So I’m going to put the rest in parentheses—that’s where we’re going to complete the square. ( x 2 6x ) 8 ( x 2 6x ) 8 size 12{ \( x rSup { size 8{2} } - 6x \) - 8} {}
Inside the parentheses, add the number you need to complete the square. Problem is, we used to add this number to both sides—but as I said before, we have no other side. So I’m going to add it inside the parentheses, and at the same time subtract it outside the parentheses, so the function is, in total, unchanged. ( x 2 6x + 9 ) 9 ̲ 8 ( x 2 6x + 9 ) 9 ̲ 8 size 12{ \( x rSup { size 8{2} } - 6x {underline {+9 \) - 9}} - 8} {}
Inside the parentheses, you now have a perfect square and can rewrite it as such. Outside the parentheses, you just have two numbers to combine. ( x 3 ) 2 17 ( x 3 ) 2 17 size 12{ \( x - 3 \) rSup { size 8{2} } - "17"} {}
Voila! You can now graph it! Vertex (3,17)(3,17) size 12{ \( 3, - "17" \) } {}opens up

Your turn!

Exercise 8

y = x 2 + 2x + 5 y = x 2 + 2x + 5 size 12{y=x rSup { size 8{2} } +2x+5} {}

  • a. Complete the square, using the process I used above, to make it y=a(xh)2+ky=a(xh)2+k size 12{y=a \( x - h \) rSup { size 8{2} } +k} {}.
  • b. Find the vertex and the direction of opening, and draw a quick sketch.

Exercise 9

x = y 2 10 y + 15 x = y 2 10 y + 15 size 12{x=y rSup { size 8{2} } - "10"y+"15"} {}

  • a. Complete the square, using the process I used above, to make it x=a(yk)2+hx=a(yk)2+h size 12{x=a \( y - k \) rSup { size 8{2} } +h} {}.
  • b. Find the vertex and the direction of opening, and draw a quick sketch.

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks