Can Plugging in (x/2) or (x - 1/2) Determine Real Zeros in a Quadratic Equation?

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Discussion Overview

The discussion revolves around determining real zeros in a quadratic equation by substituting specific expressions into the function. Participants explore the implications of different transformations and shifts of the quadratic function, focusing on how these affect the graph's intersection with the x-axis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant suggests replacing f(x) with x - x^2 and inquires about the appropriateness of substituting (x/2) and (x - 1/2) into f(x).
  • Another participant notes the range of f(x) and discusses vertical and horizontal shifts, questioning which shifts would prevent the graph from crossing the x-axis.
  • A later post reiterates the concern about shifts and states that the function y = x - x^2 - 1/2 does not intersect the line y = 0.
  • Concavity of f(x) is mentioned, prompting a discussion about how lowering the graph affects its intersection with the x-axis.
  • One participant advises writing each quadratic in standard form and checking the discriminant, indicating its relevance to understanding the nature of zeros.

Areas of Agreement / Disagreement

Participants express various viewpoints regarding the effects of transformations on the quadratic function, and no consensus is reached on the best approach to determine real zeros.

Contextual Notes

There are limitations regarding the assumptions made about the function's shifts and the specific conditions under which the discriminant is analyzed. The discussion does not resolve the implications of these transformations on the existence of real zeros.

mathland
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I can replace f(x) with x - x^2. Should I plug (x/2) into f(x)? How about (x - 1/2) into f(x)? I need the set up.

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Note the range of $f(x)$. A and B are vertical shifts, C and D are horizontal shifts. Which one of the given shifts would result in the graph of $f(x)$ not crossing the $x-\text{axis}$?
 
Greg said:
Note the range of $f(x)$. A and B are vertical shifts, C and D are horizontal shifts. Which one of the given shifts would result in the graph of $f(x)$ not crossing the $x-\text{axis}$?

Let y = the function.

y = x - x^2 - 1/2 does not cross the line y = 0.
 
Note the concavity of $f(x)$. What does that tell you about lowering the graph of $f(x)$? (the line $y=0$ does not concern us presently).
 
write each quadratic in standard form, $ax^2+bx+c$ ... check each discriminant, $D = b^2-4ac$

you know what the discriminant can tell you about the nature of zeros, right?
 

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