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)=2x²+4x−3

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1. Identify the quadratic function given: $f\left(x\right)=2x^2+4x-3$.
2. Find the vertex of the parabola using the vertex formula. The x-coordinate of the vertex is given by $x=-\frac{b}{\mathrm{2a}}$, where $a\; =\; 2$ and $b\; =\; 4$. Calculate this value to find the x-coordinate of the vertex.
3. Substitute the x-coordinate of the vertex back into the function $f(x)$ to find the y-coordinate of the vertex. This gives the vertex point $(x,\; f(x))$.
4. Determine the axis of symmetry, which is the vertical line passing through the vertex. Its equation is $x\; =$ the x-coordinate of the vertex.
5. Find the y-intercept by evaluating $f(0)$, and find the x-intercepts by solving the quadratic equation $2x^2\; +\; 4x\; -\; 3\; =\; 0$ using the quadratic formula or factoring. Use the vertex and intercepts to sketch the graph, then describe the domain (all real numbers) and the range based on the vertex's y-value and the parabola opening direction.

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) = ax^2 + bx + c. The vertex helps in sketching the graph and understanding the function's behavior.

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 is crucial for graphing the parabola and identifying symmetric points such as intercepts.

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.