In Exercises 39–44, an equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=−4x^2+8x−3

Verified step by step guidance

1. Identify the coefficient of the quadratic term in the function $f(x)\; =\; -4x^2\; +\; 8x\; -\; 3$. Here, the coefficient is $a\; =\; -4$.
2. Determine whether the parabola opens upward or downward by looking at the sign of $a$. Since $a\; =\; -4$ is negative, the parabola opens downward, so the function has a maximum value.
3. Find the x-coordinate of the vertex using the formula $x\; =\; -\backslash frac\{b\}\{2a\}$, where $b\; =\; 8$ and $a\; =\; -4$. Substitute the values to get $x\; =\; -\backslash frac\{8\}\{2(-4)\}$.
4. Calculate the maximum value by substituting the x-coordinate of the vertex back into the function: $f(x)\; =\; -4x^2\; +\; 8x\; -\; 3$. This gives the maximum value $f(x\_\{vertex\})$.
5. Identify the domain and range of the function. The domain of any quadratic function is all real numbers, $(-\backslash infty,\; \backslash infty)$. Since the parabola opens downward and has a maximum value at the vertex, the range is $(-\backslash infty,\; f(x\_\{vertex\})]$.

Key Concepts

Vertex of a Quadratic Function

The vertex of a quadratic function f(x) = ax^2 + bx + c is the point where the function attains its maximum or minimum value. It can be found using the formula x = -b/(2a). The y-coordinate of the vertex is the function's maximum or minimum value, depending on the parabola's direction.

Direction of the Parabola (Concavity)

The sign of the coefficient 'a' in a quadratic function determines whether the parabola opens upward or downward. If a > 0, the parabola opens upward and has a minimum value; if a < 0, it opens downward and has a maximum value. This helps identify whether the vertex is a minimum or maximum point.

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, the range is all y-values greater than or equal to the vertex's y-coordinate; if downward, the range is all y-values less than or equal to the vertex's y-coordinate.