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)=x²+6x+3

Verified step by step guidance

1. Identify the quadratic function: \(f(x) = x^{2} + 6x + 3\).
2. Find the vertex using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\). Here, \(a = 1\) and \(b = 6\), so calculate \(x = -\frac{6}{2 \times 1}\).
3. Substitute the x-coordinate of the vertex back into the function to find the y-coordinate: \(f\left(-\frac{6}{2}\right) = \left(-\frac{6}{2}\right)^{2} + 6 \times \left(-\frac{6}{2}\right) + 3\).
4. Find the y-intercept by evaluating \(f(0)\), which is simply the constant term in this case: \(f(0) = 3\).
5. Find the x-intercepts by solving the quadratic equation \(x^{2} + 6x + 3 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=6\), 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. It can be found using the formula x = -b/(2a) for a quadratic function f(x) = ax² + bx + c. The vertex coordinates help in sketching the graph and understanding the parabola's shape.

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 helps in graphing the parabola and identifying symmetric points on either side.

Domain and Range of a Quadratic Function

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.