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

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SUMMARY

The discussion focuses on determining real zeros in the quadratic function defined as f(x) = x - x^2. Participants explore the implications of substituting (x/2) and (x - 1/2) into f(x) and analyze the effects of vertical and horizontal shifts on the graph. It is concluded that the function y = x - x^2 - 1/2 does not intersect the x-axis, indicating no real zeros. The importance of the discriminant, D = b^2 - 4ac, in assessing the nature of zeros is emphasized.

PREREQUISITES
  • Understanding of quadratic functions and their standard form, ax^2 + bx + c
  • Knowledge of the discriminant and its role in determining the nature of zeros
  • Familiarity with vertical and horizontal shifts in graph transformations
  • Basic calculus concepts, particularly concavity and its implications
NEXT STEPS
  • Study the properties of the discriminant in depth, focusing on its implications for real zeros
  • Learn about graph transformations, specifically vertical and horizontal shifts
  • Explore concavity and its effects on the behavior of quadratic functions
  • Practice rewriting quadratic functions in standard form and analyzing their graphs
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Mathematics students, educators, and anyone interested in understanding quadratic equations and their graphical representations.

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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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