6.6 Analyzing Graphs of Quadratic Functions

Write a Quadratic Equation in Vertex form

CCSS: A.SSE.3 • CHOOSE and PRODUCE an equivalent form of an

expression to REVEAL and EXPLAIN properties of the quantity represented by the expression.

• a. FACTOR a quadratic expression to reveal the zeros of the function it defines.

• b. COMPLETE THE SQUARE iin a quadratic expression to REVEAL the maximum or minimum value of the function it defines.

CCSS: F.IF.7• GRAPH functions expressed symbolically and SHOW

key features of the graph, by hand in simple cases and using technology for more complicated cases.*

• a. GRAPH linear and quadratic functions and show intercepts, maxima, and minima.

Standards for Mathematical Practice

• 1. Make sense of problems and persevere in solving them.

• 2. Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the

reasoning of others.  • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6. Attend to precision. • 7. Look for and make use of structure. • 8. Look for and express regularity in repeated reasoning.

Essential Questions• How do I determine the domain, range, maximum,

minimum, roots, and y-intercept of a quadratic function from its graph?

• How do I use quadratic functions to model data?  • How do I solve a quadratic equation with non - real

roots?

Vertex form of the Quadratic Equation

So far the only way we seen the Quadratic Equation is ax2 + bx + c =0.

This form works great for the Quadratic Equation.

Vertex form works best for Graphing.We need to remember how to find the

vertex. The x part of the vertex come from part of the quadratic equation.

abx2

Vertex form of the Quadratic Equation

The x part of the vertex come from part of the quadratic equation.

To find the y part, we put the x part of the vertex.

The vertex as not (x, y), but (h, k)

abx2

cabb

abay

22

2

Find the vertex of the Quadratic Equation

974)1(2

71412

144

224

742

2

2

yy

y

x

xxy

Find the vertex of the Quadratic Equation

91

9,1

khVertex

The Vertex form of the Quadratic Equation

91

9,1

742 2

khVertex

xxy khxay 2

?aisWhat

The Vertex form of the Quadratic Equation

91

9,1

742 2

khVertex

xxy khxay 2

?aisWhat

2a

The Vertex form of the Quadratic Equation

91

9,1

742 2

khVertex

xxy khxay 2

2a

912

912

2

2

xy

xy

Write the Quadratic Equation in Vertex form

Find a, h and ka= 1h = -1k = 3

khxay 2

3421

4121

1122

42

2

2

yyy

x

xxy

Write the Quadratic Equation in Vertex form

Find a, h and k

a= 1h = -1k = 3

khxay 2

31

31

2

2

xy

xy

422 xxy

Vertex is better to use in graphing

y = 2(x - 3)2 – 2 Vertex (3 , -2)Put in 4 for x, y = 2(3 - 4)2 – 2 (4, 0)

Then (2, 0)is also a

point

Let see what changes happen when you change “a”

Let see what changes happen when you change “a”

Let see what changes happen when you change “a”

The larger the “a”, the skinner the graphWhat if “a” is a fraction?

Let see what changes happen when you change “a”

What if “a” is a fraction?

What if we change “h” in the Vertex

Let a = 1, k = 0

Changing the “h” moves the graph Left or Right.

What if we change “k” in the Vertex

Let a = 1, h = 0

“k” moves the graph up or down.

Write an equation

Given the vertex and a point on the graph.The vertex gives you “h” and “k”. We have to

solve for “a”Given vertex (1, 2) and point on the graph

passing through (3, 4)h =1; k = 2

2)1( 2

2

xay

khxay

Write an equation

Given vertex (1, 2) and point on the graph passing through (3, 4)

x=3, y=4

2)1( 2

2

xay

khxay

2)13(4 2 aSolve for “a”

Write an equation

a = ½

2)1( 2

2

xay

khxay

Solve for “a”

a

aaa

a

4242

244224

2)13(4

2

2

Write an equation

a = ½

2)1( 2

2

xay

khxay

Final Answer

a

aaa

a

4242

244224

2)13(42

2

2121 2 xy

6.7 Graphing and Solving Quadratic Inequalities

Solving by Graphing

Find the Vertex and the zeros of the Quadratic Equation

You can find the zero in anyway we used in this chapter. Making Table

FactoringCompleting the SquareQuadratic Formula

Solving by Graphing

Given: Zeros:

41,

23

412

233

23

23

123

23

2

2

Vertex

k

h

xxy

1;012;020120232

xxxx

xxxx

Solving by Graphing

Given: Zeros:

41,

23Vertex

12

xx

y = x 2 -3x +2

Solving by Graphing

Which way do I shade?, inside or outside

y = x 2 -3x +2

Solving by Graphing

The answers are in the shaded area

y > x2 -3x +2Why is it a

dotted line?

Solve x2 - 4x + 3 > 0

Find the zeros, (x - 3)(x – 1) = 0 x = 3 ; x = 1

Is the Graph up or Down?

Solve x2 - 4x + 3 > 0

Find the zeros, (x - 3)(x – 1) = 0 x = 3 ; x = 1

Where do we shade? Inside or Outside

Try a few points

One lower then the lowest zero, one higher then the highest zero and one in the middle.

Let x =0Let x = 4Let x = 2

Try a few points

One lower then the lowest zero, one higher then the highest zero and one in the middle.

Let x =0 02 - 4(0) + 3 > 0 TrueLet x = 4 42 – 4(4) + 3 > 0 TrueLet x = 2 22 – 4(2) + 3 > 0 FalseOnly shade where it is true.

Solve x2 - 4x + 3 > 0

Solve Quadratic Inequalities Algebraically

21

231

231

231

291

1221411

02

2

2

2

2

orx

or

xx

xx

Break the number line into three parts

Test a number less then -2 x ≤ -2

Let x = - 3(-3)2 + ( -3) ≤ 2 9 – 3 = 6

6 is not less then – 2 False

Break the number line into three parts

Test a number between -2 and 1 -2 ≤ x ≤ 1

Let x = 0(0)2 + (0) ≤ 2 0 + 0 = 0

0 is less then 2 True

Break the number line into three parts

Test a number great then 1 x ≥ 1

Let x = 2(2)2 + (2) ≤ 2 4 + 2 = 6

6 is not less then – 2 False

Break the number line into three parts

So the answer is -2 ≤ x ≤ 1