Problem 24 - Beginning and Intermediate Algebra

Question

Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ f(x)=\frac{1}{2} x^{2} $$

Step-by-Step Solution

Answer

Vertex: \((0, 0)\), Axis of Symmetry: \(x = 0\), Domain: \(( -\infty, \infty)\), Range: \([ 0, \infty )\)

Step 1 - Identify the Coefficients

The given quadratic function is \(f(x) = \frac{1}{2} x^2\). Identify the coefficients: the coefficient of \(x^2\) is \(\frac{1}{2}\), the coefficient of \(x\) is 0, and the constant term is 0.

Step 2 - Determine the Vertex

For a parabola in the form \(ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\). Here, \(a = \frac{1}{2}\) and \(b = 0\). Thus, the x-coordinate of the vertex is \(x = -\frac{0}{2 \times \frac{1}{2}} = 0\). Substituting \(x = 0\) into the function to find the y-coordinate: \(f(0) = \frac{1}{2} (0)^2 = 0\). Therefore, the vertex is \((0, 0)\).

Step 3 - Find the Axis of Symmetry

The axis of symmetry for the parabola \(y = ax^2 + bx + c\) is the vertical line that passes through the vertex. Since the vertex is \((0, 0)\), the axis of symmetry is the line \(x = 0\).

Step 4 - Determine the Domain

The domain of a quadratic function \(ax^2 + bx + c\) is all real numbers. Thus, the domain of \(f(x) = \frac{1}{2} x^2\) is \(( -\infty, \infty )\).

Step 5 - Determine the Range

Since the coefficient of \(x^2\) is positive, the parabola opens upwards. The vertex represents the minimum point. Therefore, the range of the function is all real numbers greater than or equal to 0: \([0, \infty )\).

Step 6 - Graph the Parabola

Plot the vertex \((0, 0)\) and a few additional points on either side of the vertex to sketch the parabola. For example, if \(x = 2\), then \(f(2) = \frac{1}{2} (2)^2 = 2\). If \(x = -2\), then \(f(-2) = \frac{1}{2} (-2)^2 = 2\). Plot these points along with the vertex and draw a smooth curve through them.

Step 7: Summary

To summarize: - Vertex: \((0, 0)\) - Axis of Symmetry: \(x = 0\) - Domain: \(( -\infty, \infty)\) - Range: \([ 0, \infty )\)

Key Concepts

Vertex of a Parabola

The vertex is a crucial feature for understanding parabolas in quadratic functions. In the function given, we use the standard form formula to find the vertex. For any quadratic function of the form \( ax^2 + bx + c \), the x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). Here, our function is \( f(x) = \frac{1}{2} x^2 \) where \( a = \frac{1}{2} \) , \( b = 0 \), and \( c = 0 \). Substituting these values into the formula: \[ x = -\frac{0}{2 \times \frac{1}{2}} = 0 \]. This means the x-coordinate of the vertex is 0. To find the y-coordinate, plug x back into the function: \[ f(0) = \frac{1}{2} (0)^2 = 0 \]. Therefore, the vertex of the parabola is at \( (0, 0) \). The vertex is also the minimum or maximum point of the parabola. For \( f(x) = \frac{1}{2} x^2 \), it is the minimum point because the parabola opens upwards (indicated by the positive coefficient).

Axis of Symmetry

The axis of symmetry of a parabola is the vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. It essentially mirrors the parabola on either side of this line. Given the vertex in our function \( f(x) = \frac{1}{2} x^2 \) is at \( (0, 0) \), the axis of symmetry is the line \( x = 0 \). Whenever you have a vertical parabola given by \( ax^2 + bx + c \), the axis of symmetry is \( x = -\frac{b}{2a} \). In this case, it simplifies to \( x = 0 \). This line helps you understand how the parabola is oriented and where it is centered.

Quadratic Functions Domain and Range

Understanding the domain and range of quadratic functions is key to fully grasping their behavior on a graph. The domain of a quadratic function, like \( f(x) = \frac{1}{2} x^2 \), is all real numbers. This means the function accepts any real number as an input value for x, indicating it goes infinitely in both horizontal directions. Thus, the domain is \( (-\infty, \infty) \). The range, contrastingly, is about the y-values the function can output. For \( f(x) = \frac{1}{2} x^2 \), the parabola opens upward (positive coefficient), making the vertex the lowest point. Therefore, the range of this function is y-values greater than or equal to 0, represented as \( [0, \infty) \). Knowing the domain and range helps you understand where the function exists on the coordinate plane.

Graph Sketching

Sketching a parabola involves plotting key points, starting with the vertex and incorporating the axis of symmetry. For the function \( f(x) = \frac{1}{2} x^2 \), follow these steps to graph it easily:

• Identify and plot the vertex, \( (0, 0) \).
• Draw the axis of symmetry, which is the line \( x = 0 \).
• Choose a few values for x to find corresponding y-values. For example, when \( x = 2 \): \[ f(2) = \frac{1}{2} (2)^2 = 2 \], and when \( x = -2 \): \[ f(-2) = \frac{1}{2} (-2)^2 = 2 \].
• Plot these points, \( (2, 2) \) and \( (-2, 2) \).
• Draw a smooth curve through the vertex and these points to complete the parabola.

Graph sketching helps us visualize and understand the function's behavior better. Make sure your graph reflects symmetry about the axis and accurately represents the function's curvature.