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. $f\left(x\right)=-x^2+2x+3$

Verified step by step guidance

1. 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.
2. 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 \(x = -\frac{2}{2 \times (-1)}\).
3. Substitute the x-coordinate of the vertex back into the function to find the y-coordinate: \(f\left(-\frac{b}{2a}\right) = -\left(-\frac{b}{2a}\right)^2 + 2\left(-\frac{b}{2a}\right) + 3\).
4. Find the y-intercept by evaluating \(f(0)\), which gives the point where the graph crosses the y-axis.
5. 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 \(c = 3\).

Key Concepts

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.