What is the vertex of a parabola?

Short Answer

The vertex of a parabola is the highest or lowest point on its graph. It is the point where the curve changes direction.

If the parabola opens upward, the vertex is the lowest point. If it opens downward, the vertex is the highest point. It is an important point that helps in understanding the shape of the graph.

Detailed Explanation

Vertex of parabola

A parabola is the graph of a quadratic equation, which is written as:

The vertex is a very important point on this graph. It is the point where the parabola reaches its maximum or minimum value. In simple words, it is the turning point of the curve.

Position of vertex

The position of the vertex depends on the direction in which the parabola opens:

  • If the value of a is positive, the parabola opens upward, and the vertex is the lowest point.
  • If the value of a is negative, the parabola opens downward, and the vertex is the highest point.

So, the vertex helps us understand whether the graph has a maximum value or a minimum value.

Formula to find vertex

The x-coordinate of the vertex can be found using the formula:

After finding the value of x, we substitute it into the equation to find the y-coordinate. This gives us the exact position of the vertex.

Importance of vertex

The vertex is very useful because:

  • It shows the maximum or minimum value of the function
  • It helps in graphing the parabola
  • It is used in solving real-life problems
  • It helps in understanding the shape of the curve

In many problems, especially in business and physics, we need to find the maximum profit or minimum cost, and the vertex gives this value.

Relation with axis of symmetry

The vertex lies on a vertical line called the axis of symmetry. This line divides the parabola into two equal parts.

The equation of the axis of symmetry is also:
x = −b / 2a

So, the vertex is always located on this line.

Example for understanding

Consider the equation:
y = x² − 4x + 3

Here, a = 1 and b = −4

Using the formula:
x = −(−4) / 2×1 = 4/2 = 2

Now substitute x = 2 into the equation:
y = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1

So, the vertex is (2, −1).

Real-life importance

The concept of vertex is used in many real-life situations:

  • Finding maximum profit in business
  • Finding minimum cost
  • Studying motion in physics
  • Designing structures like bridges

It helps in solving problems where we need the highest or lowest value.

Conclusion

The vertex of a parabola is the turning point of the graph where it reaches its highest or lowest value. It is a key concept in understanding quadratic equations and their graphs. By finding the vertex, we can easily analyze and solve many mathematical and real-life problems.

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