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)=2x²+4x−3

Verified step by step guidance

1. Identify the quadratic function given: \(f(x) = 2x^{2} + 4x - 3\).
2. Find the vertex of the parabola using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\), where \(a = 2\) and \(b = 4\).
3. Calculate the y-coordinate of the vertex by substituting the x-value found into the function: \(f(x) = 2x^{2} + 4x - 3\).
4. Determine the axis of symmetry, which is the vertical line passing through the vertex, given by the equation \(x = -\frac{b}{2a}\).
5. Find the x-intercepts by solving the quadratic equation \$2x^{2} + 4x - 3 = 0\( using the quadratic formula: \)x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\(, and find the y-intercept by evaluating \)f(0)$.

Key Concepts

Vertex of a Quadratic Function

The vertex of a quadratic function is the highest or lowest point on its graph, a parabola. It can be found using the formula x = -b/(2a) for a function in standard form f(x) = ax² + bx + c. The vertex helps determine the parabola's maximum or minimum value and is essential for sketching the graph accurately.

Axis of Symmetry

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

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 (a > 0), the range is all values greater than or equal to the vertex's y-coordinate; if it opens downward (a < 0), the range is all values less than or equal to the vertex's y-coordinate.