Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. See Examples 1–4. ƒ(x) = -3x^2 + 24x - 46

Verified step by step guidance

1. Rewrite the quadratic function in vertex form by completing the square: \( f(x) = -3(x^2 - 8x) - 46 \).
2. Factor out the coefficient of \( x^2 \) from the quadratic and linear terms: \( f(x) = -3(x^2 - 8x + 16 - 16) - 46 \).
3. Complete the square inside the parentheses: \( f(x) = -3((x - 4)^2 - 16) - 46 \).
4. Simplify the expression to find the vertex form: \( f(x) = -3(x - 4)^2 + 48 - 46 \).
5. Identify the vertex \((h, k)\), axis of symmetry, domain, and range from the vertex form \( f(x) = -3(x - 4)^2 + 2 \).

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, where a, b, and c are constants. 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 analyzing their characteristics.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point on its graph, depending on whether it opens downwards or upwards. For a quadratic function in standard form, the vertex can be found using the formula x = -b/(2a) to determine the x-coordinate, and then substituting this value back into the function to find the corresponding y-coordinate. The vertex is crucial for identifying the maximum or minimum value of the function.

Domain and Range

The domain of a function refers to all possible input values (x-values) that the function can accept, while the range refers to all possible output values (y-values) that the function can produce. For quadratic functions, the domain is typically all real numbers, but the range is determined by the vertex; if the parabola opens upwards, the range starts from the vertex's y-coordinate to positive infinity, and if it opens downwards, it extends from negative infinity to the vertex's y-coordinate.