Podcast
Questions and Answers
Which of the following transforms y = x^2 to the graph of y = (x + 5)^2?
Which of the following transforms y = x^2 to the graph of y = (x + 5)^2?
- A translation 5 units to the left (correct)
- A translation 5 units down
- A translation 5 units up
- A translation 5 units to the right
Which of the following transforms the graph of y = x^2 to the graph of y = x^2 - 7?
Which of the following transforms the graph of y = x^2 to the graph of y = x^2 - 7?
- Translation 7 units down (correct)
- Translation 7 units up
- Translation 7 units to the right
- Translation 7 units to the left
The graph of y = -0.2x^2 is _______________ the graph of y = x^2.
The graph of y = -0.2x^2 is _______________ the graph of y = x^2.
wider than and opens in the opposite direction of
The graph of y = 5x^2 is _______________ the graph of y = x^2.
The graph of y = 5x^2 is _______________ the graph of y = x^2.
Write the equation of the function whose graph is shown.
Write the equation of the function whose graph is shown.
To obtain the graph of y = (x - 8)^2, shift the graph of y = x^2 _________.
To obtain the graph of y = (x - 8)^2, shift the graph of y = x^2 _________.
To obtain the graph of y = x^2 - 6, shift the graph of y = x^2 _________.
To obtain the graph of y = x^2 - 6, shift the graph of y = x^2 _________.
Explain the steps you would use to determine the path of the ball given by y = -16x^2 + 32x + 3 in terms of a transformation of the graph of y = x^2.
Explain the steps you would use to determine the path of the ball given by y = -16x^2 + 32x + 3 in terms of a transformation of the graph of y = x^2.
Which of following is the graph of y = -(x + 1)^2 - 3?
Which of following is the graph of y = -(x + 1)^2 - 3?
The vertex of the graph of y = (x - 1)^2 - 5 is
The vertex of the graph of y = (x - 1)^2 - 5 is
The vertex of the graph of y = -4(x + 3)^2 + 2 is
The vertex of the graph of y = -4(x + 3)^2 + 2 is
Complete the statements below that show y = x^2 + 2x - 1 being converted to vertex form. Form a perfect-square trinomial.
Complete the statements below that show y = x^2 + 2x - 1 being converted to vertex form. Form a perfect-square trinomial.
Enter the values of h and k so that y = x^2 + 6x + 10 is in vertex form.
Enter the values of h and k so that y = x^2 + 6x + 10 is in vertex form.
Complete the statements below that show y = 8x^2 + 32x + 17 being converted to vertex form.
Complete the statements below that show y = 8x^2 + 32x + 17 being converted to vertex form.
Which of the following statements are true about the graph of f(x) = 6(x + 1)^2 - 9? Check all that apply.
Which of the following statements are true about the graph of f(x) = 6(x + 1)^2 - 9? Check all that apply.
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Study Notes
Transformations of Quadratic Functions
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Translation Example: The function y = (x + 5)² is derived from y = x² by translating the graph 5 units to the left.
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Vertical Translations: The function y = x² - 7 represents a vertical translation of the graph of y = x² downwards by 7 units.
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Wider vs. Narrower Graphs: The graph y = -0.2x² is wider than y = x² and opens in the opposite direction, while y = 5x² is narrower than y = x² and opens in the same direction.
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Graph Equation Construction: For the function graph, use y = (x + ___)² + ___, with an example being y = (x - 5)² + 3, indicating a horizontal shift to the right by 5 and a vertical shift upwards by 3.
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Horizontal and Vertical Shifts: To obtain y = (x - 8)² from y = x², shift right by 8. To convert y = x² to y = x² - 6, shift down by 6.
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Projectile Motion Representation: A ball's height, modeled by y = -16x² + 32x + 3, describes a parabolic path. Completing the square reveals a vertex form indicating a reflection across the x-axis, narrower shape, and shifts to the right by 1 unit and up by 19 units.
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Identifying Graph Type: The graph of y = -(x + 1)² - 3 is identified as the third graph option.
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Finding Vertex Coordinates: The vertex for y = (x - 1)² - 5 is at (1, -5). For y = -4(x + 3)² + 2, the vertex is at (-3, 2).
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Vertex Form Conversion Steps: Converting y = x² + 2x - 1 to vertex form involves forming a perfect-square trinomial, factoring it, and simplifying to (x + 1)² - 2.
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Finding Vertex Form Parameters: The vertex form of y = x² + 6x + 10 results in y = (x + 3)² + 1 after identifying h = -3 and k = 1.
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Converting to Vertex Form with Leading Coefficient: The function y = 8x² + 32x + 17 transforms to vertex form by first factoring out the leading coefficient, leading to y = 8(x + 2)² - 15.
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Graph Properties Analysis: For f(x) = 6(x + 1)² - 9, the true statements include that the vertex is not (1, -9), the graph opens upward, the graph is steeper than y = x², and it is equivalent to f(x) = 6x² + 12x - 3.
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