Unit 5 Review: Transformations and Quadratic Functions

Unit 5 Review: Transformations and Quadratic Functions

1. Transformations of Quadratic Function (y = x^2)

  • Given Functions:

    • a) (y = 2(x - 1)^2)

    • b) (y = -3(x - 3)^2 + 8)

  • Transformations Explained:

    • For a)

    • Vertical Stretch: The coefficient of 2 indicates a vertical stretch by a factor of 2.

    • Horizontal Shift: The expression (x - 1) indicates a horizontal shift to the right by 1 unit.

    • Vertex: The vertex of the graph, after transformations, is at ((1, 0)).

    • Graph Sketch:

    • The parabola opens upwards due to the positive leading coefficient (2).

    • Sketch includes vertex at (1, 0) and passes through points derived from substituting values into the equation.

    • For b)

    • Vertical Reflection: The negative sign indicates a reflection across the x-axis.

    • Vertical Stretch: The coefficient of -3 indicates a vertical stretch by a factor of 3.

    • Horizontal Shift: The expression (x - 3) indicates a horizontal shift to the right by 3 units.

    • Vertical Shift: The +8 indicates an upward shift by 8 units.

    • Vertex: The vertex after transformations is at ((3, 8)).

    • Graph Sketch:

    • The parabola opens downwards due to the negative leading coefficient (-3).

    • Sketch includes vertex at (3, 8) and additional points reflecting the shape of the parabola.

2. Determining Values of a and b in Quadratic Formula

  • Relation Given: (y = ax^2 + bx + 2)

  • Given Vertex: ((1, 0))

  • Substituting Vertex into the Quadratic Equation:

    • When (x = 1, y = 0):

    • (0 = a(1)^2 + b(1) + 2)

    • (0 = a + b + 2)

    • Rearranging gives:

      • (a + b = -2)

  • Finding Second Equation:

    • Assuming a second point, say another vertex related function or point on the parabola to determine values of (a) and (b).

3. Transformations Applied to (y = x^2) for Given Graph

  • List of Transformations:

    • To derive the transformations, analyze the vertex and end behavior of the derived graph.

  • Equations:

    • Write the equation in vertex form:

    • For instance, if the vertex is at ((h, k)), then it may appear as (y = a(x - h)^2 + k\) with derived values for (a), (h), and (k).

4. Determining Equation in Vertex Form and Standard Form

  • Given Point: ((-2, 2))

  • Vertex Formulation:

    • Substitute the point back into the vertex form to derive the value of (a).

  • Standard Form Conversion:

    • Convert from vertex form to standard form (y = ax^2 + bx + c) as necessary.

5. Mathematical Model for Daily Profit

  • Problem Scenario:

    • Owner's Profit based on selling price (p, d) of dresses.

    • Notable Points:

    • Maximum profit of $750 at $60 each.

    • Profit of $500 at a selling price of $45.

  • Using Quadratic Relation:

    • Assume profit function in vertex form:

    • (p(d) = a(d - 60)^2 + 750) (with corresponding transformed vertex)

    • Use given points to solve for (a) and validate the vertex form through the data provided.

6. Quadratic Relation with Zeros and Y-Intercept

  • Given Conditions:

    • Zeros at 1 and 3, Y-intercept of 3.

  • Factored Form:

    • Based on zeros, the equation can initially be stated as: (y = a(x - 1)(x - 3)).

  • Finding Value of a:

    • Substitute y-intercept into the equation to find the coefficient (a). This is where the function passes through (0, 3).

  • Converting to Standard Form:

    • Expand using algebraic methods to find standard form (y = ax^2 + bx + c).

  • Determining Vertex Form:

    • Revert the equation into vertex form after establishing necessary values.

  • Final Notes: Consider all transformations and calculations to ensure correctness in identifying parabolas with given attributes and conditions.

  1. Daily Profit Scenario: A dress shop owner notices that the maximum profit occurs when the dresses are sold for $60 each, resulting in a profit of $750. When the dresses are sold for $45 each, the profit drops to $500. Develop the profit function p(d) based on this information, and determine how many dresses need to be sold to maximize profit. Answer: A dress shop owner notices that the maximum profit occurs when the dresses are sold for $60 each, resulting in a profit of $750. When the dresses are sold for $45 each, the profit drops to $500. To develop the profit function p(d) based on this information, we can use the vertex form of a quadratic equation:

    • Assume the function is in the form:
      p(d)=a(d−60)2+750p(d)=a(d−60)2+750

    • Here, the maximum profit (vertex) occurs at a selling price of $60, and the profit is $750 at that point.

    Next, we need to determine the value of 'a'. Given $45 brings a profit of $500:

    • At point (d, p(d)) = (45, 500):
      500=a(45−60)2+750500=a(45−60)2+750

    Solving for 'a':

    1. 500=a(225)+750500=a(225)+750

    2. 500−750=225a500−750=225a

    3. −250=225a−250=225a

    4. a=−250225=−109a=−225250​=−910​

    Thus, the profit function is:
    p(d)=−109(d−60)2+750p(d)=−910​(d−60)2+750

    To determine how many dresses need to be sold to maximize profit, we find the vertex, which states that selling at 60 dresses maximizes profit. Therefore, the owner should aim to sell at this price point to achieve the highest profit.

  2. Transformation Analysis: The vertex of a quadratic function is located at (3, 8), and it opens downwards. If the quadratic function is represented in the form y = a(x - h)^2 + k, where h and k are the x and y coordinates of the vertex respectively, write a word problem that requires identifying the value of 'a' if a point on the graph is (5, 4).
    Answer: A garden designer is planning a new layout and discovers that the shape of the garden can be modeled by a quadratic function. The vertex of the garden's shape is located at the point (3, 8), indicating that its maximum height is 8 units when positioned 3 units from the origin. The garden slopes downwards as you move away from the vertex. One of the garden paths passes through the point (5, 4), which is located to the right of the vertex's x-coordinate.

    Using the vertex form of the quadratic function, which is given by the equation y = a(x - h)^2 + k, where (h, k) is the vertex (3, 8), determine the value of 'a' that describes how steep the garden's sides slope down to the path at (5, 4).

    To find 'a':
    1. Substitute the vertex (3, 8) into the equation: y = a(x - 3)^2 + 8.
    2. Use the point (5, 4) to create an equation:
    4 = a(5 - 3)^2 + 8.
    3. Solve for 'a' to determine how the path's slope affects the garden's overall shape.

  3. Zero and Y-Intercept Problem: A quadratic function has zeros at x = 1 and x = 3, and it crosses the y-axis at (0, 3). Create a word problem that asks for the standard form of the equation that represents this quadratic function and how to find the vertex of the parabola after determining the values of 'a', 'b', and 'c'.


Answer: A farmer is planting a new crop and observes that the growth of the crop can be represented by a quadratic function. He knows that the crop reaches zero growth at x = 1 and x = 3, meaning these points are the zeros of the function. Additionally, he notes that when the crop is evaluated at zero days after planting, it has a yield of 3 units at the point (0, 3).

Determine the standard form of the quadratic equation that represents the crop's growth, knowing that it can be expressed as y = a(x - 1)(x - 3).

Also, find the values of 'a', 'b', and 'c' to express the equation in standard form (y = ax^2 + bx + c). Finally, use this standard equation to determine the vertex of the parabola, which will indicate the maximum yield of the crop and the days after planting it occurs.