Problem 17 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. f(x)=-\frac{2}{5} x^{2}

Step-by-Step Solution

Answer

Vertex: (0, 0). Axis of symmetry: x = 0. Domain: (-∞, ∞). Range: (-∞, 0].

Step 1: Understand the general form

The general form of a quadratic function is given by \(f(x) = ax^2 + bx + c\). This function is a parabola.

Step 2: Identify coefficients

In the given function \(f(x) = -\frac{2}{5} x^{2}\), identify the coefficients: \(a = -\frac{2}{5}\), \(b = 0\), and \(c = 0\). Note that \(b\) and \(c\) are zero in this case.

Step 3: Find the vertex

The vertex of a parabola in standard form \(ax^2 + bx + c\) is given by the point \((h, k)\), where \(h = -\frac{b}{2a}\) and \(k = f(h)\). For this function, \(h = -\frac{0}{2 \times -\frac{2}{5}} = 0\) and \(k = f(0) = -\frac{2}{5} \times 0^2 = 0\). Therefore, the vertex is at \((0, 0)\).

Step 4: Determine the axis of symmetry

The axis of symmetry for a parabola in the form \(ax^2 + bx + c\) is given by the vertical line \(x = h\). For this function, the axis of symmetry is \(x = 0\).

Step 5: Define the domain

The domain of any quadratic function \(f(x) = ax^2 + bx + c\) is all real numbers, or \( (-\infty, +\infty) \).

Step 6: Define the range

Since the coefficient \(a\) is negative (\(a < 0\)), the parabola opens downward. Since the vertex is at \((0, 0)\), the maximum value of the function is at \(y = 0\). Therefore, the range is \( (-\infty, 0] \).

Step 7: Plot the parabola

To graph the parabola, plot the vertex at \((0, 0)\), and using additional points to the left and right of the vertex, sketch the downward opening parabola.

Key Concepts

Vertex

Graphs of quadratic functions, commonly known as parabolas, feature a crucial point called the vertex. The vertex is the peak point of the parabola, either the minimum or maximum value, depending on the direction it opens. For the function \[-\frac{2}{5} x^{2}\], the coefficients are \(a = -\frac{2}{5}\), \(b = 0\), and \(c = 0\). The vertex can be found via the formula: \[-\frac{b}{2a}\]. For our function: \[-\frac{0}{2 \times -\frac{2}{5}} = 0\]. To get the y-value, we substitute back into the function: \[-\frac{2}{5} \times 0^2 = 0\]. So the vertex is at \((0, 0)\). This point helps guide the graph's shape.

Axis of Symmetry

The axis of symmetry of a parabola is the vertical line that runs through the vertex, effectively splitting the graph into two mirror images. For any quadratic function in the form \[-\frac{2}{5} x^{2},\] it can be calculated easily. Since the vertex of our function is \( (0,0) \), the axis of symmetry will be \( x = 0 \). This vertical line is crucial in graphing because it shows that the left and right sides of the parabola are symmetrical.

Domain

The domain of a quadratic function like \[-\frac{2}{5} x^{2}\] encompasses all possible values that \(x\) can take. For parabolas, the domain is always all real numbers. In mathematical notation, the domain is expressed as \(( -\infty, +\infty)\). No matter how the parabola is positioned on the graph, there are no restrictions on \(x\)-values, making this function’s domain unrestricted and infinite.

Range

The range of a quadratic function refers to all possible values the function can output, which are the \(y\)-values. For \[-\frac{2}{5} x^{2}\], where our parabola opens downward because \( a < 0 \), the maximum value is at the vertex’s \(y\)-coordinate, which is \(0\). Thus, the range includes all values less than or equal to \(0\), expressed as \(( -\infty, 0 ]\). This informs us that the function’s outputs are restricted to a specific set of values, determined by the vertex's position and the parabola's opening direction.