Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x...
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Transcript of Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x...
![Page 1: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/1.jpg)
Warm-up• 1. Given: y = x2 – 6x + 3
• Find: Vertex, AOS, y-intercept, and graph it
• 2. Given: y = -2(x – 3)2 + 4
• Find: Vertex, AOS, y-intercept, and graph it
• 3. Given: y = 2(x – 4)(x + 2)
• Find: Vertex, AOS, y-intercept, and graph it
![Page 2: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/2.jpg)
Chapter 4
Section 4-9
Solving Quadratic Inequalities
![Page 3: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/3.jpg)
Objectives
• I can graph quadratic inequalities with the assistance of a calculator
• I can solve quadratic inequalities with a calculator or with 3 test method
![Page 4: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/4.jpg)
Review
• The solutions to any quadratic equation are the x-intercepts (where the graph crosses the x-axis)
• Now, the solutions to any quadratic inequality is where the shaded region crosses the x-axis. (It will usually be a range of values)
![Page 5: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/5.jpg)
Review from Linear Inequalities• Remember when graphing linear inequalities we
used two types of boundaries:
• Solid Line: Used to include equality. Used when equations contained ( and )
• Dashed Line: Used when equations contained ( < and >), no equality
• We will be using same ideas with quadratic graphs.
![Page 6: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/6.jpg)
Graphing Quadratic Inequalities
• To graph any quadratic inequality, you can use the following technique:
• 1. Graph the boundary (solid or dashed). Use the graphing calculator and data table to find key points.
• 2. Test a point. Point (0, 0) works best if that point is not on the boudary.
• 3. Shade the correct region based on the test results.
![Page 7: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/7.jpg)
Example 1: y x2 – 6x + 2
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
1st: Draw the boundary. Shown here in blue. It is solid because of the .
Next test a point. I chose (0, 0).
0 (0)2 – 6(0) + 2
0 0 – 0 + 2
0 2
This is true!
Since (0,0) tested good, then shade the area outside the boundary which includes (0,0).
![Page 8: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/8.jpg)
Solving by Graphingy > x2 – 9
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10 Graph boundary. Dashed because >
Test point. I chose (0,0).
0 > -9 (yes)
So shade inside boundary.So solutions are all x-values inside parabola.
Solution: (-3, 3)
Notice that –3 & 3 are not in the solution because the are on the dashed line.
![Page 9: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/9.jpg)
2nd Method: Three Test Points
• A 2nd method to solve the inequalities is by finding the roots and testing 3 regions.
• Consider the inequality below:
• x2 – x – 12 > 0 (Find solutions using 2nd Trace in calculator)
• So x = 4 or x = -3 are boundary points
![Page 10: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/10.jpg)
Solve: x2 - x – 12 > 0• x = 4 or x = -3
-3 4
Test (-5)
(-5)2 - (-5) – 12 > 0
25 + 5 – 12 > 0
30 – 12 > 0
18 > 0
YES
Test (0)
02 - (0) – 12 > 0
-12 > 0
NO
Test (5)
52 –(5) – 12 > 0
25 – 5 – 12 > 0
20 – 12 > 0
8 > 0
YES
Solutions: (-∞,-3) U (4, ∞)
![Page 11: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/11.jpg)
Solve: x2 - 8x – 33 > 0
• x2 - 8x – 33 > 0
• Use calculator to find the solutions!
• x = 11 or x = -3
• Test 3 areas x < -3, -3 < x < 11, x > 11
• Solve on next slide.
![Page 12: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/12.jpg)
Solve: x2 - 8x – 33 > 0• x = 11 or x = -3
-3 11
Test (-5)
(-5)2 - 8(-5) – 33 > 0
25 + 40 – 33 > 0
65 – 33 > 0
32 > 0
YES
Test (0)
02 - 8(0) – 33 > 0
-33 > 0
NO
Test (15)
152 –8(15) – 33 > 0
225 – 120 – 33 > 0
105 – 33 > 0
72 > 0
YES
Solutions: (-∞, -3) U (11, ∞)
![Page 13: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/13.jpg)
Solve: 2x2 - 3x - 4 0• Find solutions w/Calc.
• -.85, 2.35
-.85 2.35
Test (-1)
2(-1)2 - 3(-1) – 4 0
2 + 3 – 4 0
5 – 4 0
1 0
NO
Test (0)
2(0)2 - 3(0) – 4 0
-4 0
YES
Test (3)
2(3)2 –3(3) – 4 0
18 – 9 – 4 0
9 – 4 0
5 0
NO
Solutions: [-.85, 2.35]
![Page 14: Warm-up 1. Given: y = x 2 – 6x + 3 Find: Vertex, AOS, y-intercept, and graph it 2. Given: y = -2(x – 3) 2 + 4 Find: Vertex, AOS, y-intercept, and graph.](https://reader036.fdocuments.us/reader036/viewer/2022082822/5697bf7d1a28abf838c84b12/html5/thumbnails/14.jpg)
Homework
• WS 4-5
• Quiz Wednesday