Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x²−2x+8

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Identify the quadratic function given: $f(x) = -x^2 - 2x + 8$. This is in the standard form $f(x) = ax^2 + bx + c$ where $a = -1$, $b = -2$, and $c = 8$.

Recall that the vertex of a parabola defined by $f(x) = ax^2 + bx + c$ has an $x$-coordinate given by the formula $x = -\frac{b}{2a}$.

Substitute the values of $a$ and $b$ into the vertex formula: $x = -\frac{-2}{2 \times -1}$.

Simplify the expression to find the $x$-coordinate of the vertex.

To find the $y$-coordinate of the vertex, substitute the $x$-value back into the original function: $f(x) = -x^2 - 2x + 8$. Calculate $f(\text{x-coordinate})$ to get the $y$-coordinate.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as f(x) = ax^2 + bx + c. Its graph is a parabola, which can open upwards or downwards depending on the sign of the coefficient a.

Vertex of a Parabola

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. For f(x) = ax^2 + bx + c, the vertex's x-coordinate is found using -b/(2a), and the y-coordinate is f(-b/(2a)).

Evaluating Functions

Evaluating a function means substituting a specific value for the variable and calculating the result. To find the vertex's y-coordinate, substitute the x-value of the vertex into the original quadratic function.

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x2−2x+8