How to find the vertex of a polynomial?
Understand the Problem
The question is asking how to determine the vertex of a polynomial function, which typically involves using methods like completing the square or utilizing the vertex formula for quadratics.
Answer
The vertex is \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \).
Answer for screen readers
The vertex of the function ( f(x) = ax^2 + bx + c ) is given by: $$ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) $$
Steps to Solve
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Identify the quadratic function First, ensure that the function is in the standard quadratic form, which is ( f(x) = ax^2 + bx + c ).
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Use the vertex formula The vertex of a quadratic function can be found using the formula for the x-coordinate of the vertex: $$ x = -\frac{b}{2a} $$ where ( a ) and ( b ) are the coefficients from the quadratic function.
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Calculate the y-coordinate Once you have the x-coordinate of the vertex, substitute that value back into the original function to find the y-coordinate: $$ y = f\left(-\frac{b}{2a}\right) $$
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Write the vertex as an ordered pair Combine the x and y coordinates to express the vertex in ordered pair form: $$ \text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) $$
The vertex of the function ( f(x) = ax^2 + bx + c ) is given by: $$ \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) $$
More Information
The vertex of a quadratic function represents the maximum or minimum point of the parabola. If the parabola opens upwards (when ( a > 0 )), the vertex is a minimum point. Conversely, if the parabola opens downwards (when ( a < 0 )), the vertex is a maximum point.
Tips
- Confusing the vertex formula with that for other types of functions.
- Forgetting to substitute back into the original equation to find the y-coordinate after calculating the x-coordinate.
- Miscalculating the coefficients ( a ) or ( b ).
