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For the quadratic relation $y = -(x - 2)^2 + 9$, what is the correct direction of opening (D of O)?
For the quadratic relation $y = -(x - 2)^2 + 9$, what is the correct direction of opening (D of O)?
What is the axis of symmetry for the quadratic relation $y = -(x - 2)^2 + 9$?
What is the axis of symmetry for the quadratic relation $y = -(x - 2)^2 + 9$?
For the quadratic relation $y = -(x - 2)^2 + 9$, does it have a maximum or a minimum value, and what is that value?
For the quadratic relation $y = -(x - 2)^2 + 9$, does it have a maximum or a minimum value, and what is that value?
Which sequence represents the standard step pattern for the quadratic relation $y = -(x - 2)^2 + 9$?
Which sequence represents the standard step pattern for the quadratic relation $y = -(x - 2)^2 + 9$?
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Which transformation is represented by the negative sign in the quadratic relation $y = -(x - 2)^2 + 9$?
Which transformation is represented by the negative sign in the quadratic relation $y = -(x - 2)^2 + 9$?
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In the quadratic relation $y = -(x - 2)^2 + 9$, what does the (x - 2) term represent?
In the quadratic relation $y = -(x - 2)^2 + 9$, what does the (x - 2) term represent?
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Which transformation is represented by the + 9 in the quadratic relation $y = -(x - 2)^2 + 9$?
Which transformation is represented by the + 9 in the quadratic relation $y = -(x - 2)^2 + 9$?
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What is the vertex of the quadratic relation $y = 2(x + 7)^2 - 4$?
What is the vertex of the quadratic relation $y = 2(x + 7)^2 - 4$?
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What transformation does the step pattern 2, 6, 10 indicate?
What transformation does the step pattern 2, 6, 10 indicate?
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In the process of completing the square for $y = 2x^2 - 4x + 5$, what value is both added and subtracted within the parenthesis to maintain the equation's balance?
In the process of completing the square for $y = 2x^2 - 4x + 5$, what value is both added and subtracted within the parenthesis to maintain the equation's balance?
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Given a quadratic relation in vertex form $y = a(x - h)^2 + k$, what do $h$ and $k$ represent?
Given a quadratic relation in vertex form $y = a(x - h)^2 + k$, what do $h$ and $k$ represent?
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For the quadratic relation $y = 2(x - 1)^2 + 3$, what is the vertex?
For the quadratic relation $y = 2(x - 1)^2 + 3$, what is the vertex?
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What effect does changing the 'a' value in the quadratic equation $y = a(x-h)^2 + k$ have on the parabola?
What effect does changing the 'a' value in the quadratic equation $y = a(x-h)^2 + k$ have on the parabola?
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Which of the following expressions correctly represents the simplified form of $2(x + 1)(2x - 2)$?
Which of the following expressions correctly represents the simplified form of $2(x + 1)(2x - 2)$?
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What is the resulting expression of $y - \frac{2(2x^2 - z)}{3(x - 1)}$ after simplification, assuming no further simplification is possible without knowing the specific values of $x, y,$ and $z$?
What is the resulting expression of $y - \frac{2(2x^2 - z)}{3(x - 1)}$ after simplification, assuming no further simplification is possible without knowing the specific values of $x, y,$ and $z$?
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Given an incomplete equation $y = 3x + 6x - \text{____} + 6$, which of the following terms, when inserted into the blank space, would result in an expression that could potentially be factored?
Given an incomplete equation $y = 3x + 6x - \text{____} + 6$, which of the following terms, when inserted into the blank space, would result in an expression that could potentially be factored?
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Consider the expression: $y = \frac{2x}{4} - \frac{4}{2}$. What is an equivalent simplified form?
Consider the expression: $y = \frac{2x}{4} - \frac{4}{2}$. What is an equivalent simplified form?
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If a quadratic relation is subjected to a series of transformations, which statement is correct?
If a quadratic relation is subjected to a series of transformations, which statement is correct?
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What is the vertex of the quadratic relation represented by the equation $y = 2(x - 1)^2 + 3$?
What is the vertex of the quadratic relation represented by the equation $y = 2(x - 1)^2 + 3$?
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In the process of completing the square for the equation $y = 2x^2 - 4x + 5$, what value is added and subtracted to maintain balance?
In the process of completing the square for the equation $y = 2x^2 - 4x + 5$, what value is added and subtracted to maintain balance?
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Which transformation does a vertical stretch by a factor of 2 imply for the quadratic relation?
Which transformation does a vertical stretch by a factor of 2 imply for the quadratic relation?
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When graphing the quadratic relation $y = 3(x + 1)(x - 2)$, what is the effect of the horizontal shift indicated by the expression?
When graphing the quadratic relation $y = 3(x + 1)(x - 2)$, what is the effect of the horizontal shift indicated by the expression?
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What is the general form of a quadratic relation that involves transformations such as vertical and horizontal shifts?
What is the general form of a quadratic relation that involves transformations such as vertical and horizontal shifts?
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Which statement accurately reflects the transformation represented by a vertical shift of 4 units down in the quadratic relation?
Which statement accurately reflects the transformation represented by a vertical shift of 4 units down in the quadratic relation?
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In the expression $y = 2(x + 1)(2x - 2)$, how would you describe the direction of opening once simplified?
In the expression $y = 2(x + 1)(2x - 2)$, how would you describe the direction of opening once simplified?
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If a quadratic relation is represented as $y = ax^2 + bx + c$, what must be true about the value of 'a' in order for the quadratic to have a maximum value?
If a quadratic relation is represented as $y = ax^2 + bx + c$, what must be true about the value of 'a' in order for the quadratic to have a maximum value?
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When rewriting the quadratic relation $y = -3(x + 5)^2 + 7$, what transformation does the negative sign indicate?
When rewriting the quadratic relation $y = -3(x + 5)^2 + 7$, what transformation does the negative sign indicate?
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What does the expression $y = 3x + 6x - ext{____}$ imply for the blank space in terms of factoring potential?
What does the expression $y = 3x + 6x - ext{____}$ imply for the blank space in terms of factoring potential?
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What direction does the quadratic relation $y = 2(x + 7)^2 - 4$ open?
What direction does the quadratic relation $y = 2(x + 7)^2 - 4$ open?
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Which of the following describes the step pattern for the quadratic relation $y = -(x - 2)^2 + 9$?
Which of the following describes the step pattern for the quadratic relation $y = -(x - 2)^2 + 9$?
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In the equation $y = -(x - 2)^2 + 9$, what is the result of the transformation represented by the '2' in $(x - 2)$?
In the equation $y = -(x - 2)^2 + 9$, what is the result of the transformation represented by the '2' in $(x - 2)$?
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For the quadratic relation $y = -(x - 2)^2 + 9$, which statement is true regarding its maximum or minimum value?
For the quadratic relation $y = -(x - 2)^2 + 9$, which statement is true regarding its maximum or minimum value?
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Which transformation does the 'reflection across the x-axis' in $y = -(x - 2)^2 + 9$ indicate?
Which transformation does the 'reflection across the x-axis' in $y = -(x - 2)^2 + 9$ indicate?
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What is the minimum value of the quadratic relation $y = 2(x + 7)^2 - 4$?
What is the minimum value of the quadratic relation $y = 2(x + 7)^2 - 4$?
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Study Notes
Unit 3 - Vertex Form Test Re-Write
- Quadratic Transformations: Problems involve transforming quadratic equations using vertex form. Key transformations include vertical shifts, horizontal shifts, vertical stretches/compressions, and reflections across the x-axis.
Problem 1a
- Equation: y = (x - 2)² + 9
- Vertex: (2, 9)
- Axis of Symmetry: x = 2
- Max/Min: Maximum value of 9
- Step Pattern: 1, 3, 5 (used to graph the parabola)
- Transformations: Horizontal shift 2 units right, vertical shift 9 units up.
Problem 1b
- Equation: y = 2(x + 7)² – 4
- Vertex: (-7, -4)
- Axis of Symmetry: x = -7
- Max/Min: Minimum value of -4
- Step Pattern: 2, 6, 10
- Transformations: Vertical stretch by a factor of 2, horizontal shift 7 units left, vertical shift 4 units down.
Completing the Square
- Problem 2: Involves converting a quadratic equation from standard form to vertex form by "completing the square."
- Example: y = 2x² - 4x + 5 transforms to y = 2(x - 1)² + 3.
- Example Vertex: (1, 3)
- Key steps: The process involves grouping terms with 'x' together, factoring out the coefficient of the x² term, completing the square within the parenthesis, and simplifying the resulting equation to vertex form.
Changing Forms
- Problem 3: Involves converting equations between standard and vertex form.
- Example: Part a) Changing y = 3(x + 1)² - 2 to standard form results in y = 3x² + 6x + 4. Part b) involves a similar process given y = 2(x + 1)(2x - 2), which simplifies to y = 4x² - 4.
- Key steps: Expanding factored expressions, combining like terms and aligning with standard form are crucial.
Spicy Vertex Form
- Problem 4: Given transformations, determine the quadratic equation in vertex form.
- Example: Vertical reflection across the x-axis, vertical stretch by a factor of 4, horizontal shift 5 units right, vertical shift 2 units up results in the quadratic relation: y = -4(x - 5)² + 2.
- Key takeaways: Identifying the transformations helps determine the coefficients and constants in the vertex form equation.
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Description
This quiz focuses on understanding quadratic transformations and how to convert equations to vertex form. Participants will solve problems involving vertical shifts, horizontal shifts, stretches, and reflections. Additionally, there is a section dedicated to completing the square for converting standard form to vertex form.
