Determine the equation of the parabola

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SUMMARY

The equation of the parabola with a range of y|y≧-6 and x-intercepts at -5 and 3 is expressed as $y = k(x+5)(x-3)$, where $k$ is a positive constant. The vertex of the parabola is located at $(x,-6)$, with the x-coordinate of the vertex being the midpoint between the x-intercepts, calculated as $x = \frac{-5 + 3}{2} = -1$. By substituting the vertex coordinates into the equation, the value of $k$ can be determined, ensuring that the parabola opens upwards to meet the specified range.

PREREQUISITES
  • Understanding of quadratic equations and their standard forms
  • Knowledge of vertex form of a parabola
  • Ability to calculate midpoints between two points
  • Familiarity with the concept of positive and negative constants in equations
NEXT STEPS
  • Learn how to derive the vertex form of a parabola from its intercepts
  • Study the implications of the range on the orientation of parabolas
  • Explore the effects of varying the constant $k$ on the shape of the parabola
  • Practice solving for constants in quadratic equations using vertex coordinates
USEFUL FOR

Students studying algebra, mathematics educators, and anyone interested in understanding the properties of parabolas and their equations.

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Determine the equation of the parabola with range y|y≧-6 and x-intercepts at -5 and 3.
 
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$y = k(x+5)(x-3)$, where $k$ is a constant

$y \ge -6 \implies k > 0$

vertex is located at $(x,-6)$ where $x$ is midway between the two x-intercepts

determine the x-value of the vertex, then use the vertex coordinates to determine the value of $k$
 

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