Questions and Answers
What is Project 1 about?
Exploring translations of polynomials.
What causes a horizontal translation in the function f(x) = (x − h)^2 + k?
Changing the value of h.
What causes a vertical translation in the function f(x) = (x − h)^2 + k?
Changing the value of k.
What happens to the function f(x) = x^2 when it becomes f(x) = x^2 - 3?
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Which change in the function rule will translate f(x) = x^3 to the right?
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Which change in the function rule will translate f(x) = x^4 to the left?
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What happens to the function f(x) = x^4 when it becomes f'(x) = x^4 + 5?
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What is Project 2 about?
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What causes a horizontal dilation in the dilation rule f'(x) = af(1/bx)^2?
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What causes a vertical dilation in the dilation rule f'(x) = af(1/bx)^2?
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What happens to the function f(x) = (x - 1)^2 + 1 when it becomes f'(x) = 2((x - 1)^2 + 1)?
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What happens to the function f(x) = (x - 1)^2 + 1 when it becomes f'(x) = (1/3x - 1)^2 + 1?
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What happens to the function f(x) = (x - 1)^2 + 1 when it becomes f'(x) = 1/2((x - 1)^2 + 1)?
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What happens to the function f(x) = (x - 1)^2 + 1 when it becomes f'(x) = (2x - 1)^2 + 1?
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Which transformation reflects f(x) = (x - 1)^2 + 1 over the x-axis?
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Which transformation reflects f(x) = (x - 1)^2 + 1 over the y-axis?
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What is Project 3 about?
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Calculate the quotient: $x^3 - 7x + 14 ÷ x + 4$.
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Calculate the quotient: $5x^7 − 18x^5 ÷ x^2 − 2$.
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Enter the division problem as you would in the CAS: $9x^4 + 2x^3 ÷ x + 10$.
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What is the quotient of the problem that Johnathan entered?
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What did Sarah do wrong when entering the division problem in the CAS?
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What is Project 4 about?
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Study Notes
Project 1: Exploring Translations of Polynomials
- Horizontal translation in quadratic functions occurs by changing the value of h in the equation f(x) = (x − h)² + k.
- Vertical translation occurs by altering the value of k in the same equation.
- The function f(x) = x² transitions 3 units downwards when adjusted to f(x) = x² - 3.
- Translating f(x) = x³ to the right is achieved through the rule f'(x) = (x - 8)³.
- To translate f(x) = x⁴ to the left, use the rule f'(x) = (x + 9)⁴.
- Modifying f(x) = x⁴ to f'(x) = x⁴ + 5 results in a movement of 5 units upward.
Project 2: Exploring Dilations of Polynomials
- Horizontal dilation in quadratic functions is controlled by adjusting the value of b in the rule f'(x) = af(1/bx)².
- Vertical dilation is influenced by changing the value of a in the same dilation rule.
- The function f(x) = (x - 1)² + 1 transforms to f'(x) = 2((x - 1)² + 1), resulting in stretching away from the x-axis.
- Changing f(x) to f'(x) = (1/3x - 1)² + 1 causes stretching away from the y-axis.
- The function f(x) = (x - 1)² + 1 becomes compressed towards the x-axis when changed to f'(x) = (1/2)((x - 1)² + 1).
- Changing f(x) to f'(x) = (2x - 1)² + 1 results in stretching away from the y-axis.
- Reflecting the graph of f(x) over the x-axis can be achieved with f'(x) = -1.4((x - 1)² + 1).
- Reflection over the y-axis is accomplished by using the transformation f'(x) = (0.1x - 1)² + 1.
Project 3: Dividing Polynomials with Technology
- To find the quotient of x³ - 7x + 14 divided by x + 4, the result is x² - 4x + 9 - 22/(x + 4).
- For the division of 5x⁷ − 18x⁵ by x² − 2, the quotient is 5x⁵ - 8x³ - 16x - 32x/(x² - 2).
- When dividing the expression 9x⁴ + 2x³ by x + 10, it should be entered in a CAS tool as Division(9x⁴ + 2x³, x + 10).
- The quotient from Johnathan's division problem entered into the CAS yields x² - 11x + 60 - 358/(x + 6).
- Sarah incorrectly entered the division by using "divide" instead of "division" in the CAS.
Project 4
- Information for this project is currently missing or incomplete.
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Description
This quiz explores the concepts of translating and dilating polynomial functions, specifically focusing on quadratic, cubic, and quartic equations. Learn how changes in parameters affect the graphs of these functions. Master these transformations and dilations to enhance your understanding of polynomial behavior.
