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)=x^2+3x−10

Verified step by step guidance

1. <Step 1: Identify the coefficients in the quadratic function f(x) = x^2 + 3x - 10. Here, a = 1, b = 3, and c = -10.>
2. <Step 2: Find the vertex of the parabola using the formula for the x-coordinate of the vertex, x = -b/(2a).>
3. <Step 3: Substitute the x-coordinate of the vertex back into the function f(x) to find the y-coordinate of the vertex.>
4. <Step 4: Determine the axis of symmetry, which is the vertical line that passes through the vertex. The equation is x = -b/(2a).>
5. <Step 5: Identify the x-intercepts by setting f(x) = 0 and solving the quadratic equation x^2 + 3x - 10 = 0. Also, find the y-intercept by evaluating f(0).>

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the general shape and properties of parabolas is essential for sketching their graphs.

Vertex and Axis of Symmetry

The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. For a quadratic function in standard form, the vertex can be found using the formula x = -b/(2a), which also helps in determining the axis of symmetry.

Domain and Range

The domain of a function refers to all possible input values (x-values) for which the function is defined, while the range refers to all possible output values (y-values). For quadratic functions, the domain is typically all real numbers, but the range depends on the vertex's position. If the parabola opens upwards, the range starts from the y-coordinate of the vertex to positive infinity; if it opens downwards, it extends from negative infinity to the y-coordinate of the vertex.