Pregunta
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Graph the parabola and find its vertex and the axis of the parabola.

Ask by Ross Delgado. in the United States
Jan 30,2025

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The vertex of the parabola is at , and the axis of symmetry is .

Solución

To graph the parabola and determine its vertex and axis of symmetry, follow these steps:

1. Identify the Vertex and Axis of Symmetry

The general form of a quadratic function is:
For the given parabola:
a. Finding the Vertex:
The vertex of a parabola in the form can be found by completing the square.
Step-by-Step:
  1. Start with the original equation:
  2. Complete the square for the quadratic part:
    • Take the coefficient of , which is -6.
    • Take half of it: .
    • Square it: .
  3. Rewrite the equation by adding and subtracting this square inside the equation:
  4. Identify the vertex:
    So, the vertex is .
b. Finding the Axis of Symmetry:
The axis of symmetry is the vertical line that passes through the vertex. Its equation is:
So, the axis of symmetry is .

2. Graphing the Parabola

While I can’t provide a visual graph here, I can outline key points to help you plot the parabola:
Vertex:
Axis of Symmetry:
Finding Additional Points:
Choose x-values around the vertex and calculate the corresponding y-values.
  • For :
    Point:
  • For :
    Point:
  • For :
    Point:
  • For :
    Point:
  • For :
    Point:
  • For :
    Point:
Plotting these points along with the vertex and drawing a symmetric U-shaped curve will give you the graph of the parabola.

3. Summary

  • Vertex:
  • Axis of Symmetry:
  • Additional Points to Plot:
    • , ,
    • , ,
By plotting these points and ensuring symmetry around the axis , you’ll accurately graph the parabola .

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Extra Insights

To graph the parabola described by the equation , we first complete the square. Rewriting the equation gives us . Completing the square on yields . So, the equation can be expressed as . This shows that the vertex of the parabola is at the point and opens upward. The axis of symmetry is the vertical line .
Another fun fact about parabolas is that they are not just mathematical constructs; they can be found in physics! The arc of a ball thrown in the air can be modeled by a parabolic equation like the one above. This means you could use these mathematical concepts to predict how high and how far your throw will go—just like a real-life athlete!
If you’re interested in pushing your understanding further, look into the quadratic formula and how it derives from completing the square. It’s a powerful tool not just for finding the roots of equations, but also for understanding the properties and graphing of quadratics. These insights can give you a fresh perspective on how math applies to various fields such as engineering and computer graphics.

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