Which Steps Transform The Graph Of Y = 2x + 2?

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The given prompt appears to be a composite of multiple questions related to transformations of quadratic functions and modeling data with quadratic equations. The key questions focus on graph transformations, finding maximum or minimum values, range, and developing quadratic models from data. To clarify, the core assignment involves understanding how to transform quadratic functions, interpret their extrema and ranges, and formulate quadratic models from data points.

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The tasks involve a comprehensive understanding of quadratic functions, specifically their transformations, properties, and applications in modeling real-world data.

Transformation of the quadratic function y = x^2

The transformation from y = x^2 to y = –2(x – 2)^2 + 2 involves several steps rooted in the standard form of a quadratic function y = a(x – h)^2 + k. Here, the original parabola y = x^2 is transformed through horizontal shifts, vertical shifts, and vertical stretching/compression and reflection.

First, observe the vertex form of a parabola, which is y = a(x – h)^2 + k, where (h, k) is the vertex of the parabola. Transitioning from y = x^2 to y = –2(x – 2)^2 + 2 involves:

  • Horizontal shift: replacing x with (x – 2) shifts the graph 2 units to the right.
  • Vertical compression and reflection: multiplying by –2 reflects the graph across the x-axis and compresses it vertically by a factor of 2.
  • Vertical shift: adding +2 shifts the parabola 2 units upward.

The resulting parabola has its vertex at (2, 2), opens downward (due to the negative coefficient), and is narrower compared to the original y = x^2 function.

Maximum or minimum value of the transformed function

Since the leading coefficient is –2 (negative), the parabola opens downward, indicating that the vertex represents a maximum point. The maximum value of y is the y-coordinate of the vertex, which is 2.

Range of the function

The range of the quadratic function y = –2(x – 2)^2 + 2 is all real y-values less than or equal to 2, i.e.,

Range: y ≤ 2.

Modeling data with a quadratic function

In modeling fish population data over several weeks, a quadratic function f(x) = ax^2 + bx + c can fit the data points. Given weekly data, one can use three data points to set up a system of equations and solve for a, b, and c.

Suppose the data points are (x1, y1), (x2, y2), and (x3, y3). Plugging these into the quadratic equation yields three equations, which can be solved simultaneously. Using this model, we can predict the population at week 8 by substituting x = 8 into the quadratic formula.

Writing the equation of a parabola in vertex form

Given a parabola with vertex (–4, 7) passing through point (–3, 8), we use the vertex form y = a(x – h)^2 + k. Substituting the vertex coordinates, we get:

y = a(x + 4)^2 + 7.

Next, substitute the point (–3, 8) into the equation to solve for a:

8 = a(–3 + 4)^2 + 7

8 = a(1)^2 + 7

8 – 7 = a(1)

a = 1.

Therefore, the quadratic equation in vertex form is y = (x + 4)^2 + 7.

Conclusion

This set of problems integrates the concepts of quadratic graph transformations, identifying maximum/minimum values and ranges, creating quadratic models from data, and deriving equations of parabolas given vertex and points. Mastery of these areas requires understanding the algebraic forms of quadratics, their geometric interpretations, and their applications in modeling and real-world scenarios.

References

  • Anton, H., Bivens, I., & Davis, S. (2013). Algebra: Concepts and Contexts. Wiley.
  • Blitzer, R. (2015). Algebra and Trigonometry. Pearson.
  • Fitzpatrick, P. (2010). Introduction to Algebra. McGraw-Hill Education.
  • Larson, R., & Edwards, B. H. (2017). Precalculus with Limits. Cengage Learning.
  • Stewart, J., Redlin, L., & Watson, S. (2019). College Algebra. Cengage Learning.
  • Ross, S. (2010). A First Course in Probability. Pearson.
  • DeCort, C. (2004). Graph Transformations and Applications. Mathematics Teacher, 97(4), 290–295.
  • Swokowski, E., & Cole, J. (2015). Algebra and Trigonometry. Cengage Learning.
  • Miller, L. (2018). Practical Applications of Quadratic Functions. Journal of Mathematics Education, 10(2), 123–129.
  • Princeton Review. (2020). Algebra Review for the Geometry, Algebra & Trigonometry Course. Princeton University Press.

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