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Algebra 1

9.10.4 Rewriting Expressions in Vertex Form

Algebra 19.10.4 Rewriting Expressions in Vertex Form

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Activity

1. Here is one way to rewrite 3x2+12x+93x2+12x+9 in vertex form (of a quadratic expression).

a. Study the steps and write a brief explanation of what is happening at each step.

Original expression 3x2+12x+93x2+12x+9

Step 1 - 3(x2+4x+3)3(x2+4x+3)

Step 2 - 3(x2+4x+4+34)3(x2+4x+4+34)

Step 3 - 3(x2+4x+41)3(x2+4x+41)

Step 4 - 3((x+2)21)3((x+2)21)

Step 5 - 3(x+2)233(x+2)23

b. What is the xx-coordinate of the vertex (of a graph) that represents this expression?

c. What is the yy-coordinate of the vertex of the graph that represents this expression?

d. Does the graph open upward or downward? Explain how you know.

2. Rewrite 2x24x+62x24x+6 in vertex form.

3. Rewrite 4x2+24x+204x2+24x+20 in vertex form.

4. Rewrite x2+20xx2+20x in vertex form.

Are you ready for more?

Extending Your Thinking

1. Fill in the missing parts of the function f(x)=2(x3)(x9)f(x)=2(x3)(x9) in vertex form f(x)=a.(xb.c.)2d.e.f(x)=a.(xb.c.)2d.e. without completing the square. (Hint: Think about finding the zeros of the function.)

a. What is the value of a.

b. What is the sign of b.

Multiple Choice:

Subtraction(-)

Addition(+)

c. What is the value of c.

d. What is the sign of d.

Multiple Choice:

Subtraction(-)

Addition(+)

e. What is the value of e.

f. Be prepared to show your reasoning for your answers.

2. Fill in the missing parts of the function g(x)=2(x3)(x9)+21g(x)=2(x3)(x9)+21 in vertex form g(x)=a.(xb.c.)2d.e.g(x)=a.(xb.c.)2d.e. without completing the square.

a. What is the value of a.

b. What is the sign of b.

Multiple Choice:

Subtraction(-)

Addition(+)

c. What is the value of c.

d. What is the sign of d.

Multiple Choice:

Subtraction(-)

Addition(+)

e. What is the value of e.

f. Be prepared to show your reasoning for your answers.

Video: Rewriting Expressions in Vertex Form

Watch the following video to learn more about rewriting expressions in vertex form.

Self Check

Write 2 x 2 + 20 x + 20 in vertex form.
  1. 2 ( x + 5 ) 2 15
  2. 2 ( x 2 + 10 x + 10 )
  3. 2 ( x + 5 ) 2 30
  4. 2 ( x + 5 ) 2

Additional Resources

Vertex Form with a Leading Coefficient Besides 1

Write 2x2+8x+42x2+8x+4 in vertex form (of a quadratic expression).

Original expression 2x2+8x+42x2+8x+4

Step 1 - Rewrite the expression with 2 as a factor. 2(x2+4x+2)2(x2+4x+2)

Step 2 - Add 4 to complete the square. Subtract 4 to keep the value of the expression the same. 2(x2+4x+4+24)2(x2+4x+4+24)

Step 3 - Combine like terms so we can see the perfect square, x2+4x+4x2+4x+4. 2(x2+4x+42)2(x2+4x+42)

Step 4 - Rewrite x2+4x+4x2+4x+4 as a squared expression. 2((x+2)22)2((x+2)22)

Step 5 - Apply the distributive property to put the expression in vertex form. 2(x+2)242(x+2)24

Try it

Try It: Vertex Form with a Leading Coefficient Besides 1

Write 4x2+24x+124x2+24x+12 in vertex form.

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