How to find the maximum height of a quadratic equation?

Understand the Problem

The question is asking how to determine the maximum height of a quadratic equation, which involves identifying the vertex of the quadratic function represented in standard form. The maximum height occurs at the vertex for a downward-opening parabola.

Answer

The maximum height occurs at the vertex $ y = f\left(-\frac{b}{2a}\right) $.

Steps to Solve

Identify the quadratic function

The quadratic function is usually in the form ( f(x) = ax^2 + bx + c ). Identify the coefficients ( a ), ( b ), and ( c ) from your given equation.

Determine the vertex's x-coordinate

For a quadratic equation in standard form, the x-coordinate of the vertex can be found using the formula:

$$ x = -\frac{b}{2a} $$

Plug in the values of ( a ) and ( b ) from your function to calculate ( x ).

Calculate the maximum height (vertex y-coordinate)

To find the maximum height, substitute the value of ( x ) back into the quadratic function. This will give you the y-coordinate of the vertex.

$$ y = f(x) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c $$

Express the final maximum height

The value of ( y ) calculated from the previous step represents the maximum height of the quadratic function.

The maximum height of the quadratic function occurs at the vertex and can be calculated as ( f\left(-\frac{b}{2a}\right) ).

More Information

The maximum height of a quadratic function is important in various real-world applications, such as physics (for projectile motion) and engineering (for designing parabolic structures). The vertex represents the peak or trough of the graph depending on whether the parabola opens up or down.

Tips

Forgetting to check if the parabola opens upwards or downwards, as this determines whether the vertex is a maximum or minimum.
Miscalculating the values of ( a ), ( b ), and ( c ) from the given quadratic function.

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