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Converting from Standard Form to Vertex Form

9.10.3 Converting from Standard Form to Vertex Form

9.10.3 • Converting from Standard Form to Vertex Form

Activity

You've expanded from factored form to standard form. To learn how to convert from standard form to vertex form (of a quadratic expression), start by going backward and convert a quadratic from vertex form to standard form.

1. Rewrite (x+5)2+1(x+5)2+1 in standard form.

Compare your answer:

x2+10x+26x2+10x+26

2. Rewrite (x−3)2−7(x−3)2−7 in standard form.

Compare your answer:

x2−6x+2x2−6x+2

3. Think about the steps you took, and about reversing them. Try converting one or both of the expressions in standard form back into vertex form. Explain how you go about converting the expressions.

Compare your answer:

To convert x2−6x+2x2−6x+2 back to vertex form, you can complete the square by adding 9 and then subtracting 9 so that the expression remains unchanged. The part that is a perfect square, x2−6x+9x2−6x+9, can be written as (x−3)2(x−3)2, and then there is the 2−92−9, which equals -7.

4. Test your strategy by rewriting x2+10x+9x2+10x+9 in vertex form.

Compare your answer:

(x+5)2−16(x+5)2−16

5. Let's check the expression you rewrote in vertex form. Use the Desmos graphing tool or technology outside the course to graph both x2+10x+9x2+10x+9 and your new expression.

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a. Does it appear that both expressions define the same function?

Yes. Both expressions seem to produce the same graph.

b. If you convert your expression in vertex form back into standard form, do you get x2+10x+9x2+10x+9?

Compare your answer:

(x+5)2−16=x2+10x+25−16(x+5)2−16=x2+10x+25−16, which is x2+10x+9x2+10x+9.

Self Check

Which of the following is the equivalent vertex form of the expression x 2 + 14 x − 3 ?
  1. ( x + 7 ) 2 + 46
  2. ( x + 7 ) 2 − 52
  3. ( x + 7 ) 2 − 3
  4. ( x + 7 ) 2

Additional Resources

Standard to Vertex Form

Different forms are useful for finding different characteristics of quadratic functions.

Practice writing x2+8x−7x2+8x−7 into vertex form (of a quadratic expression).

Step 1 - Work to complete the square:

b=8b=8, so (b2)2=(82)2=16(b2)2=(82)2=16.

Step 2 - Add the value to create the perfect square, and subtract the value from the original constant (this is like adding 0).

x2+8x+16−7−16x2+8x+16−7−16

Step 3 - Factor the trinomial and simplify.

(x+4)2−23(x+4)2−23

Try it

Standard to Vertex Form

Convert the quadratic expression x2+16x−10x2+16x−10 to vertex form.

Here is how to write the quadratic expression in vertex form:

Step 1 - Work to complete the square:

b=16b=16, so (b2)2=(162)2=64(b2)2=(162)2=64.

Step 2 - Add the value to create the perfect square, and subtract the value from the original constant (this is like adding 0).

x2+16x+64−10−64x2+16x+64−10−64

Step 3 - Factor the trinomial and simplify.

(x+8)2−74(x+8)2−74