Algebra 2 Chapter 4
Quadratic Functions and Relations
4.1
Characteristics
of Quadratic Functions
IXL Practice
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K.3 Complete a function table: quadratic functions Q9X
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K.1 Characteristics of quadratic functions: graphs WMS
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K.2 Characteristics of quadratic functions: equations L8C
Main Ideas
By using the quadratic equation you can calculate the equation for the axis of symmetry, y-intercept, the coordinate for the vertex, and the maximum or minimum value.
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The y-intercept is the x-value
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The formula for the axis of symmetry is x = -b/(2a)
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To find the vertex take the value of the axis of symmetry and plug it back into the quadratic equation. Write your answer as an ordered pair (x, y)
4.2
Solve Quadratic Equations by Graphing
IXL Practice
Main Ideas
Graph the equation & the solutions/roots/zeros are where the equation goes through the x-axis. Most commonly there are two solutions, but sometimes there is only one solution or no solution.
You can graph these on the graphing calculator by pressing "2nd, trace, 2" and then doing the left bound and right bound. It's also super simple to graph them on Desmos and just click on the point for the x-axis.
4.3
Solve Quadratic Equations by Factoring
IXL Practice
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Example 1
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Example 2, 3, 4
Main Ideas
We can factor quadratic equations to solve them so we don't always have to graph them to solve them.
Types of Factoring
4.4
Complex Numbers
IXL Practice
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Example 1:
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Example 2:
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Example 5:
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Example 2 & 6:
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Example 7:
Main Ideas
How are imaginary numbers useful?
Facts to Know
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The square root of -1 is i
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i squared is -1
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Simplifying Complex Numbers Video
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Adding and Subtracting Complex Numbers Video
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Multiplying Complex Numbers Video
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Dividing Complex Numbers Video
4.5
Completing the Square
IXL Practice
Main Ideas
We can solve these equations for x by using the "completing the square" method.
You are starting with a quadratic equation in the format of
ax^2 + bx + c = 0 (make sure a equals 1, if not divide by that number first)
1. If your c value is not a perfect square, add/subtract it to the other side
ax^2 + bx + ____ = c + ____
2. Now you need to find the value to go in those blanks that will make it a perfect square trinomial. Take half of b and square it. Add it to both sides to keep the equation balanced.
(1/2b)^2
ax^2 + bx + (1/2b)^2 = c + (1/2b)^2
3. Factor the perfect square trinomial using the factoring pattern (square root the first, square root the last, make sure b is doubled)
( x + (1/2b))^2 = c + (1/2b)^2
4. Square root both sides to cancel out the square. Make sure you use a +- symbol in front of the number on the right side when you square root it.
5. Solve for x by adding/subtracting the # to the other side. You will always have two answers because of the +- when you took the square root.
4.6
The Quadratic Formula
IXL Practice
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Example 1, 2, 3, 4
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Example 5
Main Ideas
Memorize the quadratic formula and be familiar with how the quadratic equation connects to the last section on completing the square. This video shows the proof of how the quadratic formula works as a shortcut to completing the square.
Understand how the discriminate of the equation can actually tell you a lot about the type of answer you will get.
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a positive perfect square will get you two real rational solutions
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a positive non-perfect square will get you two irrational solutions
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zero will get you one real solution
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a negative number will give you two complex solutions