In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $f (x) = - x^{2} + 2 x + 3$

검증된 단계별 안내

Identify the quadratic function: $f(x) = -x^2 + 2x + 3$. Notice it is a parabola opening downward because the coefficient of $x^2$ is negative.

Find the vertex using the formula for the x-coordinate of the vertex: $x = -\frac{b}{2a}$. Here, $a = -1$ and $b = 2$, so calculate $x = -\frac{2}{2 \times (-1)}$.

Substitute the x-coordinate of the vertex back into the function to find the y-coordinate: $f\left(-\frac{b}{2a}\right) = -\left(-\frac{b}{2a}\right)^2 + 2\left(-\frac{b}{2a}\right) + 3$.

Find the y-intercept by evaluating $f(0)$, which gives the point where the graph crosses the y-axis.

Find the x-intercepts by solving the quadratic equation $-x^2 + 2x + 3 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = -1$, $b = 2$, and $c = 3$.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

주요 개념

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

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex's x-coordinate is found using -b/(2a). The vertex helps in sketching the graph and determining the range.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is x = -b/(2a). This line is crucial for graphing and understanding the parabola's symmetry.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the parabola's direction: if it opens downward (a < 0), the range is all values less than or equal to the vertex's y-coordinate; if upward (a > 0), it is all values greater than or equal to the vertex's y-coordinate.

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=−x2+2x+3f(x) = -x^2 +2x + 3