In Exercises 17–38, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x−x^2+3

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1. Rewrite the quadratic function in standard form by rearranging the terms: $f\left(x\right)=-x^2+2x+3$.
2. Find the vertex of the parabola using the formula for the x-coordinate of the vertex: $x=-b/\mathrm{2a}$, where $a\; =\; -1$ and $b\; =\; 2$.
3. Calculate the y-coordinate of the vertex by substituting the x-value found into the function $f(x)$.
4. Determine the axis of symmetry, which is the vertical line passing through the vertex, given by $x\; =$ the x-coordinate of the vertex.
5. Identify the domain and range: the domain of any quadratic function is all real numbers, and the range depends on whether the parabola opens up or down (since $a\; =\; -1$, it opens downward, so the range is all values less than or equal to the y-coordinate of the vertex).

Key Concepts

Quadratic Functions and Their Graphs

A quadratic function is a polynomial of degree two, typically written as f(x) = ax^2 + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and orientation of the parabola is essential for sketching its graph.

Vertex and Axis of Symmetry

The vertex is the highest or lowest point on the parabola, found using the formula x = -b/(2a). The axis of symmetry is a vertical line through the vertex that divides the parabola into two mirror images. Knowing the vertex and axis helps in accurately sketching the graph and identifying key features.

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 vertex: if the parabola opens upward, the range is all y-values greater than or equal to the vertex's y-coordinate; if downward, all y-values less than or equal to the vertex's y-coordinate. This helps describe the function's output values.