Quadratics Unit 3 Vertex Form Test

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Questions and Answers

For the quadratic relation $y = -(x - 2)^2 + 9$, what is the correct direction of opening (D of O)?

  • Left
  • Up
  • Right
  • Down (correct)

What is the axis of symmetry for the quadratic relation $y = -(x - 2)^2 + 9$?

  • x = 2 (correct)
  • y = 9
  • x = -2
  • y = -9

For the quadratic relation $y = -(x - 2)^2 + 9$, does it have a maximum or a minimum value, and what is that value?

  • Maximum of 2
  • Maximum of 9 (correct)
  • Minimum of 9
  • Minimum of 2

Which sequence represents the standard step pattern for the quadratic relation $y = -(x - 2)^2 + 9$?

<p>1, 3, 5 (C)</p> Signup and view all the answers

Which transformation is represented by the negative sign in the quadratic relation $y = -(x - 2)^2 + 9$?

<p>Reflection across the x-axis (C)</p> Signup and view all the answers

In the quadratic relation $y = -(x - 2)^2 + 9$, what does the (x - 2) term represent?

<p>Horizontal shift 2 units to the right (D)</p> Signup and view all the answers

Which transformation is represented by the + 9 in the quadratic relation $y = -(x - 2)^2 + 9$?

<p>Vertical shift 9 units up (B)</p> Signup and view all the answers

What is the vertex of the quadratic relation $y = 2(x + 7)^2 - 4$?

<p>(-7, -4) (A)</p> Signup and view all the answers

What transformation does the step pattern 2, 6, 10 indicate?

<p>Vertical stretch by a factor of 2. (B)</p> Signup and view all the answers

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?

<p>$1$ (A)</p> Signup and view all the answers

Given a quadratic relation in vertex form $y = a(x - h)^2 + k$, what do $h$ and $k$ represent?

<p>h represents the x-coordinate of the vertex, and k represents the y-coordinate of the vertex. (B)</p> Signup and view all the answers

For the quadratic relation $y = 2(x - 1)^2 + 3$, what is the vertex?

<p>$(1, 3)$ (C)</p> Signup and view all the answers

What effect does changing the 'a' value in the quadratic equation $y = a(x-h)^2 + k$ have on the parabola?

<p>It affects the direction of opening and vertical stretch/compression of the parabola. (A)</p> Signup and view all the answers

Which of the following expressions correctly represents the simplified form of $2(x + 1)(2x - 2)$?

<p>$4x^2 - 4$ (B)</p> Signup and view all the answers

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 $z$?

<p>$y - \frac{4x^2 - 2z}{3x - 3}$ (A)</p> Signup and view all the answers

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?

<p>$3x^2$ (C)</p> Signup and view all the answers

Consider the expression: $y = \frac{2x}{4} - \frac{4}{2}$. What is an equivalent simplified form?

<p>$y = \frac{x}{2} - 2$ (B)</p> Signup and view all the answers

If a quadratic relation is subjected to a series of transformations, which statement is correct?

<p>The vertex form can be directly determined from the transformations. (C)</p> Signup and view all the answers

What is the vertex of the quadratic relation represented by the equation $y = 2(x - 1)^2 + 3$?

<p>(1, 3) (A)</p> Signup and view all the answers

In the process of completing the square for the equation $y = 2x^2 - 4x + 5$, what value is added and subtracted to maintain balance?

<p>4 (A)</p> Signup and view all the answers

Which transformation does a vertical stretch by a factor of 2 imply for the quadratic relation?

<p>The parabola opens narrower. (C)</p> Signup and view all the answers

When graphing the quadratic relation $y = 3(x + 1)(x - 2)$, what is the effect of the horizontal shift indicated by the expression?

<p>Shifted left by 1 unit. (C)</p> Signup and view all the answers

What is the general form of a quadratic relation that involves transformations such as vertical and horizontal shifts?

<p>y = a(x - h)^2 + k (A)</p> Signup and view all the answers

Which statement accurately reflects the transformation represented by a vertical shift of 4 units down in the quadratic relation?

<p>The vertex is lowered by 4 units. (A)</p> Signup and view all the answers

In the expression $y = 2(x + 1)(2x - 2)$, how would you describe the direction of opening once simplified?

<p>It opens upwards. (C)</p> Signup and view all the answers

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?

<p>a must be less than 0. (B)</p> Signup and view all the answers

When rewriting the quadratic relation $y = -3(x + 5)^2 + 7$, what transformation does the negative sign indicate?

<p>Reflection across the x-axis. (C)</p> Signup and view all the answers

What does the expression $y = 3x + 6x - ext{____}$ imply for the blank space in terms of factoring potential?

<p>It should balance the equation. (B)</p> Signup and view all the answers

What direction does the quadratic relation $y = 2(x + 7)^2 - 4$ open?

<p>Upward (A)</p> Signup and view all the answers

Which of the following describes the step pattern for the quadratic relation $y = -(x - 2)^2 + 9$?

<p>1, 3, 5 (B)</p> Signup and view all the answers

In the equation $y = -(x - 2)^2 + 9$, what is the result of the transformation represented by the '2' in $(x - 2)$?

<p>Horizontal shift 2 units right (D)</p> Signup and view all the answers

For the quadratic relation $y = -(x - 2)^2 + 9$, which statement is true regarding its maximum or minimum value?

<p>It has a maximum value of 9 (D)</p> Signup and view all the answers

Which transformation does the 'reflection across the x-axis' in $y = -(x - 2)^2 + 9$ indicate?

<p>The graph flips over the x-axis (A)</p> Signup and view all the answers

What is the minimum value of the quadratic relation $y = 2(x + 7)^2 - 4$?

<p>-4 (B)</p> Signup and view all the answers

Flashcards

Vertex

The point where the parabola changes direction. It's also the highest or lowest point on the graph.

Axis of Symmetry

The vertical line that passes through the vertex of a parabola. It divides the parabola into two symmetrical halves.

Direction of Opening (D of O)

Indicates whether the parabola opens upwards (minimum) or downwards (maximum).

Maximum or Minimum

The value of y at the vertex. It represents the maximum or minimum value of the quadratic function.

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Step Pattern

A pattern that helps us sketch a parabola. In the standard form, the step pattern is 1, 3, 5.

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Vertical Reflection

A transformation that flips the graph across the x-axis. It changes the direction of opening.

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Horizontal Shift

A transformation that shifts the graph horizontally. A positive value shifts the graph to the right, a negative value shifts it to the left.

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Vertical Shift

A transformation that shifts the graph vertically. A positive value shifts the graph upwards, a negative value shifts it downwards.

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Completing the Square

A technique used to rewrite a quadratic expression in the form y=a(x-h)^2 + k, where (h,k) is the vertex of the parabola. This form allows for easy identification of the vertex and other key features of the parabola.

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Standard Form of Quadratic

A form of a quadratic expression written as y = ax^2 + bx + c. It is the most common and general form for quadratic equations.

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Vertex Form

A form of writing a quadratic equation where the vertex is clearly visible (h, k) and the 'a' value controls the direction and width of the parabola. The formula is y = a(x - h)^2 + k, where (h, k) represents the vertex and 'a' represents the vertical stretch or compression.

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Converting to Vertex Form

The process of rewriting a quadratic equation from standard form (ax^2 + bx + c) to vertex form (y = a(x - h)^2 + k). This involves completing the square.

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Coefficient 'a' in Vertex Form

The number that multiplies the (x - h)^2 term in the vertex form equation. It controls the direction and width of the parabola.

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Horizontal Shift (Vertex Form)

A transformation that moves the graph horizontally. A positive value shifts the graph to the right, a negative value shifts it to the left.

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Simplifying Expressions

The process of simplifying a quadratic expression by combining like terms and applying the order of operations.

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What is the step pattern in the standard form of a quadratic equation?

In the standard form of a quadratic equation (y = ax^2 + bx + c), the step pattern helps you sketch the parabola by moving up/down in specific increments.

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What does a vertical reflection do to a parabola?

This transformation flips the parabola over the x-axis. If the original parabola opened upwards, it now opens downwards, and vice-versa.

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What is a horizontal shift in a parabola?

This transformation shifts the parabola left or right along the x-axis. A positive value shifts it to the right, and a negative value shifts it to the left.

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What is a vertical shift in a parabola?

This transformation shifts the parabola up or down along the y-axis. A positive value shifts it upwards, and a negative value shifts it downwards.

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What is the vertex form of a quadratic equation?

This form is written as y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. It's useful for easily identifying the vertex.

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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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