​

A-SSE.1a

​Interpret parts of an expression, such as terms, factors, and coefficients. 

​Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

KNOW: ​complete the square to rewrite quadratic equations from standard form to graphing form.

SHOW: determine the vertex of a parabola given a quadratic function in graphing form.

ACTIVITY 1: ​RED LIGHT , GREEN LIGHT

​Complete the square to rewrite in graphing form and identify the vertex of parabola.

​1. f(x) = x² + 5x + 2

2. ​f(x) = x² + 8x + 15

3. ​f(x) = x² – 4x​

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EXAMPLE 1: ​f(x) = x² + 5x + 2

Solution:

Step 1: ​f(x) – 2 = x² + 5x ( Move the constant term to other equation.)

Step 2: ​x² + 5x = (5/2)² ( Identify what needs to be added to both sides of 25/4 the equation into a perfect square by taking half

x term coefficient and​ squaring it.)

Step 3: ​ f (x)− 2 + 25/4 = x² + 5x + 25/ 4 (Add 25/4 to both sides of the equation )

Step 4: f (x)+ 17/4 =( x + 5 /2 )² (Simplify the left side and factor the trinomial on the right side. )

Step 5: f (x) =( x + 5/2 )² − 17/4 (Graphing Form of f(x) = (x – h)² + k )

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The vertex is (− 5/2 , − 17/4 ) or (–2.5, –4.25) ​

​The vertex is

− 5/2 , − 17 /4 or

(–2.5, –4.25).

The y-intercept is where x = 0. Substitute x = 0 into the original equation: f(0) = 0² + 5(0) + 2 = 2; the y-intercept is (0, 2)

​Graphing Form of f(x) = (x – h)² + k

Example 2:f(x) = x² + 8x + 15

Step 1: ​f(x)-15=x² +8x+___

Step 2: (8/2)² =(4)²​ or 16

Step 3: ​f(x)-15+16=x² +8x+16

Step4: ​f(x)+1= (x+4)²

​Step 5: f(x)=(x+4)² -1

The vertex is​ is at (h, k) = (–4, –1)

Y-intercept=(0,15)​

​Graphing Form of f(x) = (x – h)² + k

Example 3:f(x) = x² – 4x​

Step 1:f(x)+0=x²-4x+___​

Step 2: ​ (-4/2)² =(-2)² or 4

Step3: f(x)+0+4=x²-4x+4​

Step 4: f(x)+4=​( x-2)²

​Step 5: f(x)=(x-2)² -4

​The vertex is at (h,k) =(2, –4)

​Complete the square to rewrite the equation of each function in graphing form. Then state the vertex of the parabola.

​1. f(x) = x² + 6x + 7

2. f(x) = x² + 4x + 11

3. f(x) = x ² + 10x

4. f(x) = x² + 7x + 2

5. f(x) = x² – 6x + 9