Finding the Minimum Value of a Parabola: Tips and Techniques

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SUMMARY

The minimum value of the quadratic function f(x) = 3x² + 10x + 219 can be determined using the vertex formula, where the x-coordinate of the vertex is calculated as x = -b/(2a). For this function, a = 3 and b = 10, leading to x = -10/(2*3) = -5/3. Substituting x = -5/3 back into the function yields the minimum value of f(-5/3) = 3(-5/3)² + 10(-5/3) + 219, which simplifies to 204.33.

PREREQUISITES
  • Understanding of quadratic functions and their standard form.
  • Familiarity with the vertex formula for parabolas.
  • Basic algebra skills for function evaluation.
  • Knowledge of function transformations and graphing techniques.
NEXT STEPS
  • Learn how to derive the vertex of different types of quadratic functions.
  • Explore the implications of the vertex on the graph of a parabola.
  • Study the effects of changing coefficients a, b, and c on the shape and position of parabolas.
  • Investigate optimization techniques for quadratic functions in real-world applications.
USEFUL FOR

Students studying algebra, mathematicians interested in optimization, and educators teaching quadratic functions and their properties.

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What is the smallest value in the image set of the function:

f(x)=3x^2+10x+219
 
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Welcome to the forum!

The vertex (minimum or maximum) of a parabola $f(x)=ax^2+bx+c$ has $x$-coordinate $x=-\dfrac{b}{2a}$.

For the future, please read the forum http://mathhelpboards.com/rules/, especially rule #11. In this case, you could write what facts about parabolas you know and what exactly your difficulty is in applying them.
 

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