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4.12 Graphing quadratic functions ii

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Further problems on graphing quadratic functions.

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

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.
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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.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

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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 ( y − k ) 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.

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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 ( x − k ) 2 + h size 12{y=a \( x - k \) rSup { size 8{2} } +h} {} . They looked more like…well, you know, x 2 − 2x − 8 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!

Step Example
The function itself 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 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 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 size 12{ \( x - 3 \) rSup { size 8{2} } - "17"} {}
Voila! You can now graph it! Vertex ( 3, − 17 ) size 12{ \( 3, - "17" \) } {} opens up

Your turn!

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 ( x − h ) 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.
Got questions? Get instant answers now!

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 ( y − k ) 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.
Got questions? Get instant answers now!
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Source:  OpenStax, Advanced algebra ii: activities and homework. OpenStax CNX. Sep 15, 2009 Download for free at http://cnx.org/content/col10686/1.5
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