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Questions and Answers
Which quadratic function has a graph that is shifted right 4, up 2, and opens down compared to the original $y = x^2$?
What is the vertex of the graph of $y = -2x^2 + 8x - 3$?
Which of the following is a solution to the equation $x^2 - 6x + 5 = 0$ when solved by factoring?
What are the roots of the equation $3x^2 - 9x = 0$ after factoring?
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What is the result of simplifying $(-10 + 12i) - (4 - 7i)$?
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What is the simplified form of $(-2i)(3i)(-4i)(5i)$?
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Which of the following is the result of simplifying $i^{35}$?
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Simplifying the expression $4 - 20$ results in which of the following?
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What are the values of m and n if (12 - m) + 20i = 4 - 5ni?
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What is the axis of symmetry for the quadratic function y = 4x^2 + 3?
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What are the roots of the quadratic equation x^2 - 8x + 15 = 0 when solved by factoring?
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What is the equation of the quadratic when its roots are -1 and 2?
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Which form represents the completed square of the equation x^2 - 4x + 5 = 0?
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What is the vertex of the quadratic function in the form y = 2x^2 - 4x + 4?
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What is the direction of opening for the quadratic equation y = 4x^2 + 3?
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What is the result when using the quadratic formula on x^2 + 8x + 13 = 0?
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Study Notes
Quadratic Functions and Their Properties
- Quadratic functions can be transformed by shifting (right, left, up, down) and reflecting over the axis.
- Standard form for a quadratic function is ( y = ax^2 + bx + c ).
- Vertex form is given by ( y = a(x - h)^2 + k ), where (h, k) is the vertex and a indicates direction (up or down).
Vertex and Axis of Symmetry
- The vertex of a quadratic function can be found using the formula ( h = -\frac{b}{2a} ).
- The axis of symmetry is a vertical line given by ( x = h ).
- For ( y = ax^2 + bx + c ), if ( a < 0 ), the parabola opens downward; if ( a > 0 ), it opens upward.
Solving Quadratic Equations
- Quadratic equations can be solved by:
- Factoring
- Completing the square
- Using the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )
- Roots can also be deduced from the factored form ( (x - r_1)(x - r_2) = 0 ) where ( r_1 ) and ( r_2 ) are the roots.
Complex Numbers
- Simplifying expressions involving imaginary numbers must maintain the form of ( a + bi ).
- The imaginary unit ( i ) is defined as ( i^2 = -1 ).
- Multiplying complex numbers involves the distributive property, treating ( i ) as a variable.
Simplification Strategies
- To simplify expressions, factor common terms when possible.
- Use conjugates for division to eliminate imaginary parts in the denominator.
- Combine like terms and ensure answers are presented in standard form ( a + bi ).
Practice Problems
- Problems include determining the vertex and axis of symmetry, solving quadratics by various methods, and working with complex numbers through simplification and multiplication.
- Examples often involve equations that need to be factored or reduced, emphasizing the importance of showing work for full understanding.
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Description
Assess your understanding of quadratic functions in Algebra II Chapter 3. This quiz covers transformations of quadratic graphs, including shifts and reflections. Test your ability to identify correct equations based on given transformations.
