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This an Algebra 2 lesson Introducing solving quadratic inequalities. Students have learned extensively about quadratics and should bring a little prior knowledge concerning basic concepts of the inequality relation. This lesson uses two classrooms and measuring tape.
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Warm-up 1
What is the area of Mr. Paul’s room in square meters?
Warm-up 2: key words and symbols used in inequalities
• Greater than
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Solving Quadratic Inequalities
What it means to be less than a function
FYI: Aligned Common Core State Standards
CCSS: Mathematics, CCSS: HS: Algebra, Creating Equations
HSA-CED.A. Create equations that describe numbers or relationships
HSA-CED.A.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Modeling with quadratic functions
Certain events are accurately modeled by quadratic functions. For example:
•Area•Projectiles (objects shot up into the air which are
pulled down by gravity)•Profit and loss
Area
Ms. Ilonka wants her room to be increased in size by at least 20 square meters. She wants to increase the length and the width by the same amount. To model this, we need to know the dimensions of her room and then figure out by how much we need to increase the length and width.
Area – from blue to red
Area – how many square metersis Ms. Ilonka’s room?
Area – how many square metersis Ms. Ilonka’s room?
Area - modeling the situation
The function that describes the area of a rectangle is a quadratic function
Area – “at least”
We can write a quadratic equation to show the area of a rectangle
We can write a quadratic inequality to show if the area must be at least 20 square meters more than the original area
Projectile
Throw a ball forward and up and observe the shape it makes. How do you think we can model this with a quadratic curve?
Projectile
Let 𝑓 𝑥 = −𝑥2 + 4𝑥 + 5 be the model for throwing a ball. The independent variable is time in seconds and the dependent variable is vertical distance in feet. During what time will the ball be at least 8 ft high?
Projectile
Let 𝑓 𝑥 = −𝑥2 + 4𝑥 + 5 be the model for throwing a ball. The independent variable is time in seconds and the dependent variable is vertical distance in feet. During what time will the ball be at least 8 ft high?
Step 1: graph the function
Projectile
Let 𝑓 𝑥 = −𝑥2 + 4𝑥 + 5 be the model for throwing a ball. The independent variable is time in seconds and the dependent variable is vertical distance in feet. During what time will the ball be at least 8 ft high?
Step 1: graph the function (scale and label)
Step 2: draw a horizontal line at 8 ft
Projectile
Let 𝑓 𝑥 = −𝑥2 + 4𝑥 + 5 be the model for throwing a ball. The independent variable is time in seconds and the dependent variable is vertical distance in feet. During what time will the ball be at least 8 ft high?
Step 1: graph the function (scale and label)
Step 2: draw a horizontal line at 8 ft
Step 3: Draw two vertical lines where your horizontal line intersected the quadratic curve
Projectile
Let 𝑓 𝑥 = −𝑥2 + 4𝑥 + 5 be the model for throwing a ball. The independent variable is time in seconds and the dependent variable is vertical distance in feet. During what time will the ball be at least 8 ft high?
Step 1: graph the function (scale and label)Step 2: draw a horizontal line at 8 ftStep 3: Draw two vertical lines where your horizontal line intersected the quadratic curveStep 4: write an inequality based on the x-values (time interval in seconds)
Class time practice
Complete the problems from the textbook
Write the problem, each step and the solution in your notebook
p.114 #18-23
Then some word problems
p.114 # 11 & p.115 #47
Out of class practice - homework
p.116 #52, 53, 56, 62-65