1. Chapter 6.6 Analyzing Graphs of Quadratic Functions Standard & Honors
Algebra II Mr. Gilbert
Chapter 6.6
Analyzing Graphs of Quadratic Functions
Standard & Honors
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2. Agenda Warm up Homework Review Lesson New Homework Check your answers
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3. Warm-up 6.6 Complete the square: Problem 1: x2 –12x = 28
Problem 2: ax2 –bx +c =0
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4. (sing the quadratic formula)
Warm-up 6.6
Complete the square:
Problem 1: x2 –12x = 28
x2 -12x +62 = > (x-6)2 = 82
x-6 = +-8
Ans. {x | x=-2,14}
Problem 2: ax2 –bx +c =0
(sing the quadratic formula)
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5. Homework Review
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6. 6.6 Analyzing Graphs of Quadratic Functions
Example 1 Graph a Quadratic Function in Vertex Form (3)
Example 2 Write y = x2 + bx + c in Vertex Form (3)
Example 3 Write y = ax2 + bx + c in Vertex Form, a  1 (3)
Example 4 Write an Equation Given Points (4)
Recall the standard form of the quadratic equation
y= ax2 + bx +c
The vertex form of the quadratic equation
y= a(x-h)2 + k
Question: What does the vertex form show more quickly than the standard form?
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Lesson 6 Contents

7. h = 3 and k = 2 Analyze Then draw its graph.
Answer: The vertex is at (h, k) or (3, 2) and the axis of symmetry is The graph has the same shape as the graph of but is translated 3 units right and 2 units up.
Now use this information to draw the graph.
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Example 6-1a

8. Step 2 Draw the axis of symmetry,
Step 1 Plot the vertex, (3, 2).
Step 2 Draw the axis of symmetry,
(1, 6)
(5, 6)
(2, 3)
(4, 3)
Step 3 Find and plot two points on one side of the axis of symmetry, such as (2, 3) and (1, 6).
(3, 2)
Step 4 Use symmetry to complete the graph.
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Example 6-1a

9. Analyze Then draw its graph.
Answer:
The vertex is at (–2, –4), and the axis of symmetry is The graph has the same shape as the graph of ; it is translated 2 units left and 4 units down.
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Example 6-1b

10. Write in vertex form. Then analyze the function.
Notice that is not a perfect square.
Balance this addition by subtracting 1.
Complete the square by
adding
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Example 6-2a

11. Write as a perfect square.
This function can be rewritten as So, and
Answer: The vertex is at (–1, 3), and the axis of symmetry is Since the graph opens up and has the same shape as but is translated 1 unit left and 3 units up.
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Example 6-2a

12. Write in vertex form. Then analyze the function.
Answer: vertex: (–3, –4); axis of symmetry: opens up; the graph has the same shape as the graph of but it is translated 3 units left and 4 units down.
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Example 6-2b

13. Write in vertex form. Then analyze and graph the function.
Original equation
Group and factor, dividing by a.
Complete the square by adding 1 inside the parentheses.
Balance this addition by subtracting –2(1).
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Example 6-3a

14. Write as a perfect square.
Answer: The vertex form is So, and The vertex is at (–1, 4) and the axis of symmetry is Since the graph opens down and is narrower than It is also translated 1 unit left and 4 units up.
Now graph the function. Two points on the graph to the right of are (0, 2) and (0.5, –0.5). Use symmetry to complete the graph.
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Example 6-3a

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Example 6-3a

16. Write in vertex form. Then analyze and graph the function.
Answer:
vertex: (–1, 7); axis of symmetry: x = –1; opens down; the graph is narrower than the graph of y = x2,and it is translated 1 unit left and 7 units up.
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Example 6-3b

17. Substitute 1 for h, 2 for k, 3 for x, and 4 for y.
Write an equation for the parabola whose vertex is at (1, 2) and passes through (3, 4).
The vertex of the parabola is at (1, 2) so and Since (3, 4) is a point on the graph of the parabola , and Substitute these values into the vertex form of the equation and solve for a.
Vertex form
Substitute 1 for h, 2 for k, 3 for x, and 4 for y.
Simplify.
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Example 6-4a

18. Subtract 2 from each side.
Divide each side by 4.
Answer: The equation of the parabola in vertex form is
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Example 6-4a

19. Check A graph of verifies that the
parabola passes through the point at (3, 4).
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Example 6-4a

20. Write an equation for the parabola whose vertex is at (2, 3) and passes through (–2, 1).
Answer:
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Example 6-4b

21. Homework - Honors See Syllabus 6.5
Pg multiples of 3, 40-45
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22. Homework
See Syllabus 6.5
Pg multiples of 3
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