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−x²+3

Verified step by step guidance

1. Rewrite the quadratic function in standard form by rearranging the terms: \(f(x) = -x^2 + 2x + 3\).
2. Find the vertex using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\), where \(a = -1\) and \(b = 2\) in the equation \(f(x) = ax^2 + bx + c\).
3. Calculate the y-coordinate of the vertex by substituting the x-value found into the function: \(f(x) = -x^2 + 2x + 3\).
4. Determine the axis of symmetry, which is the vertical line passing through the vertex, given by the equation \(x = \) (the x-coordinate of the vertex).
5. Find the x-intercepts by setting \(f(x) = 0\) and solving the quadratic equation \(-x^2 + 2x + 3 = 0\), and find the y-intercept by evaluating \(f(0)\).

Key Concepts

Quadratic Functions and Parabolas

A quadratic function is a polynomial of degree two, typically written as f(x) = ax^2 + bx + c. Its graph is a parabola, which can open upward or downward depending on the sign of 'a'. Understanding the shape and properties of parabolas is essential for graphing and analyzing quadratic functions.

Vertex and Axis of Symmetry

The vertex of a parabola is its highest or lowest point, 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. Identifying the vertex and axis helps in sketching the graph and understanding its 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 vertex: if the parabola opens upward, the range is all values greater than or equal to the vertex's y-coordinate; if downward, all values less than or equal to it. Determining domain and range is key to understanding the function's output values.