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

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1. Identify the quadratic function given: $f\left(x\right)=x^2+6x+3$.
2. Find the vertex of the parabola by using the vertex formula for the x-coordinate: $x=-\frac{b}{\mathrm{2a}}$, where $a\; =\; 1$ and $b\; =\; 6$. Then substitute this x-value back into the function to find the y-coordinate of the vertex.
3. Determine the axis of symmetry, which is the vertical line that passes through the vertex. Its equation is $x\; =$ the x-coordinate of the vertex found in the previous step.
4. Find the y-intercept by evaluating $f(0)$. This gives the point where the parabola crosses the y-axis.
5. Find the x-intercepts (if any) by solving the quadratic equation $x^2\; +\; 6x\; +\; 3\; =\; 0$ using the quadratic formula: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$. These intercepts are the points where the parabola crosses the x-axis.
6. Use the vertex and intercepts to sketch the graph, noting that the parabola opens upward since $a\; =\; 1\; >\; 0$.
7. Determine the domain of the function, which for any quadratic function is all real numbers: $(-\backslash infty,\; \backslash infty)$.
8. Determine the range of the function based on the vertex. Since the parabola opens upward, the range is all values greater than or equal to the y-coordinate of the vertex.

Key Concepts

Vertex of a Quadratic Function

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

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). This line is crucial for graphing because it shows the parabola's symmetry and helps locate points on the graph.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since the parabola extends infinitely left and right. 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.