Warm Up Linear Relationships Sudoku

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Homework Questions

Chapter 1 - Sections 5 through 8 Main Ideas Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 Graphing a Quadratic Function Quadratic Models Complex Numbers Key Terms Imaginary Unit i Complex Number Complex Conjugates Quadratic Equation Completing the Square Discriminant Quadratic Function Quadratic Model

𝟐 𝒙 𝟐 −𝟕𝒙−𝟒=𝟎 𝟐𝒙+𝟏 𝒙−𝟒 =𝟎 𝟐𝒙+𝟏=𝟎 𝒐𝒓 𝒙−𝟒=𝟎 𝒙=− 𝟏 𝟐 𝒐𝒓 𝒙=𝟒 Main Ideas Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 Use factoring if a, b, and c are integers and 𝑏 2 −4𝑎𝑐 is a perfect square. 𝟐 𝒙 𝟐 −𝟕𝒙−𝟒=𝟎 𝟐𝒙+𝟏 𝒙−𝟒 =𝟎 𝟐𝒙+𝟏=𝟎 𝒐𝒓 𝒙−𝟒=𝟎 𝒙=− 𝟏 𝟐 𝒐𝒓 𝒙=𝟒

𝒙 𝟐 +𝟒𝒙+𝟏=𝟎 𝒙 𝟐 +𝟒𝒙=−𝟏 𝒙 𝟐 +𝟒𝒙+ 𝟒 𝟐 𝟐 =−𝟏+ 𝟒 𝟐 𝟐 𝒙+𝟐 𝟐 =𝟑 Main Ideas Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 Solve by completing the square if a = 1 and b is even. 𝒙 𝟐 +𝟒𝒙+𝟏=𝟎 𝒙 𝟐 +𝟒𝒙=−𝟏 𝒙 𝟐 +𝟒𝒙+ 𝟒 𝟐 𝟐 =−𝟏+ 𝟒 𝟐 𝟐 𝒙+𝟐 𝟐 =𝟑

𝒙+𝟐 𝟐 =𝟑 𝒙+𝟐=± 𝟑 𝒙=−𝟐+ 𝟑 𝒐𝒓 𝒙=−𝟐− 𝟑 Main Ideas Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 Solve by completing the square if a = 1 and b is even. Continued 𝒙+𝟐 𝟐 =𝟑 𝒙+𝟐=± 𝟑 𝒙=−𝟐+ 𝟑 𝒐𝒓 𝒙=−𝟐− 𝟑

Use the quadratic formula 𝒙= −𝒃 ± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂 otherwise. Main Ideas Solving a Quadratic Equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 Use the quadratic formula 𝒙= −𝒃 ± 𝒃 𝟐 −𝟒𝒂𝒄 𝟐𝒂 otherwise. Online Notes & Practice Solving Quadratics

𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄 Main Ideas Graphing a Quadratic Function If 𝑎>0, the graph is ∪shaped If 𝑎<0, the graph is ∩shaped The point 0,𝑐 is on the graph. −𝑏+ 𝑏 2 −4𝑎𝑐 2𝑎 ,0 and −𝑏 − 𝑏 2 −4𝑎𝑐 2𝑎 ,0 are on the graph. The equation of the axis of symmetry is x=− 𝑏 2𝑎 The vertex has x-coordinate − 𝑏 2𝑎

𝒚= 𝒂 𝒙−𝒉 𝟐 +𝒌 If 𝑎>0, the graph is ∪shaped Main Ideas Graphing a Quadratic Function 𝒚= 𝒂 𝒙−𝒉 𝟐 +𝒌 If 𝑎>0, the graph is ∪shaped If 𝑎<0, the graph is ∩shaped The vertex is ℎ,𝑘 . The axis is 𝑥=ℎ.

If you have a quadratic model, Main Ideas Quadratic Models If you have a quadratic model, you can use the model to predict data values (see Example 2 on page 45) or to maximize or minimize the function (evaluate f when 𝑥=− 𝑏 2𝑎 ).

Key Terms Discriminant The value 𝑏 2 −4𝑎𝑐 for a quadratic equation 𝑎 𝑥 2 +𝑏𝑥+𝑐=0 If 𝑏 2 −4𝑎𝑐>0, there are two real roots. If 𝑏 2 −4𝑎𝑐=0, there is one real root. If 𝑏 2 −4𝑎𝑐<0, there are two complex roots.

Quadratic Review Stations Work through each station and solve as many problems as you can. Write down the problems in your notebook!

=𝒊 𝟐𝟕 −𝒊 𝟑 −𝟐𝟕 − −𝟑 = 𝟑 𝟑 − 𝟑 𝒊 =𝟐𝒊 𝟑 𝟓−𝟕𝒊 − −𝟐+𝒊 =𝟓−𝟕𝒊+𝟐−𝒊 =𝟕−𝟖𝒊 Main Ideas Complex Numbers To simplify an expression that contains the square root of a negative number, begin by writing the expression in terms of i. −𝟐𝟕 − −𝟑 =𝒊 𝟐𝟕 −𝒊 𝟑 = 𝟑 𝟑 − 𝟑 𝒊 =𝟐𝒊 𝟑 To add or subtract complex numbers, group the real parts and the imaginary parts. 𝟓−𝟕𝒊 − −𝟐+𝒊 =𝟓−𝟕𝒊+𝟐−𝒊 =𝟕−𝟖𝒊

Main Ideas Complex Numbers To multiply two complex numbers use FOIL (first, outer, inner, last) and substitute −1 for 𝑖 2 . 𝟐+𝟑𝒊 𝟓−𝒊 =𝟏𝟎−𝟐𝒊+𝟏𝟓𝒊−𝟑 𝒊 𝟐 =𝟏𝟎+𝟏𝟑𝒊−𝟑 −𝟏 =𝟏𝟑+𝟏𝟑𝒊

Main Ideas Complex Numbers To simplify an expression with a complex number in the denominator, multiply the numerator and denominator by the complex conjugate of the denominator. 𝟏 𝟏+𝒊 𝟑 = 𝟏 𝟏+𝒊 𝟑 ∗ 𝟏−𝒊 𝟑 𝟏−𝒊 𝟑 = 𝟏−𝒊 𝟑 𝟏−𝟑 𝒊 𝟐 = 𝟏−𝒊 𝟑 𝟏−𝟑 −𝟏 = 𝟏 𝟒 − 𝟑 𝟒 𝒊 Online Notes & Practice for Complex Numbers

Worksheet for Sections 1.5 through 1.8 Homework Worksheet for Sections 1.5 through 1.8