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)=5x²−5x

Verified step by step guidance

1. Identify the coefficient of the quadratic term in the function $f\left(x\right)=5x^2-5x$. Since the coefficient of $x^2$ is positive (5), the parabola opens upward, which means the function has a minimum value.
2. To find the minimum value and where it occurs, use the vertex formula for the x-coordinate of the vertex: $x=-b/\mathrm{2a}$, where $a\; =\; 5$ and $b\; =\; -5$ from the quadratic function $f(x)\; =\; ax^2\; +\; bx\; +\; c$.
3. Calculate the x-coordinate of the vertex by substituting the values: $x=-(-5)\; /\; (2\; *5)$. This gives the x-value where the minimum occurs.
4. Find the minimum value of the function by substituting the x-coordinate of the vertex back into the original function: $f(x)\; =\; 5x^2\; -\; 5x$. This will give the minimum value of the function.
5. Determine the domain and range of the function. The domain of any quadratic function is all real numbers, $(-\infty ,\; \infty )$. Since the parabola opens upward and has a minimum value, the range is all real numbers greater than or equal to the minimum value found in the previous step.

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 minimum or maximum value. It can be found using the formula x = -b/(2a). The y-coordinate of the vertex is the function's minimum or maximum value, depending on the parabola's direction.

Direction of the Parabola (Concavity)

The sign of the coefficient 'a' in f(x) = ax^2 + bx + c determines whether the parabola opens upward or downward. If a > 0, the parabola opens upward, and the vertex is a minimum point. If a < 0, it opens downward, and the vertex is a 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, all y-values less than or equal to the vertex's y-coordinate.