Problem 19 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=\frac{1}{2} x^{2}-5\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{1}{2} x^{2}-5\) is a parabola opening upwards with a vertex at (0, -5).

Step 1: Figure out the form of the quadratic function

The given function \(f(x)=\frac{1}{2} x^{2}-5\) is a quadratic function. It is in the standard form of a quadratic function which is \(f(x) = ax^{2} + bx + c\), where 'a' is the coefficient of \(x^{2}\), 'b' is the coefficient of x and 'c' is a constant. In this case, \(a = \frac{1}{2}\), 'b' is zero and \(c = -5\).

Step 2: Identify the vertex of the parabola

The vertex of a parabola given in this form is at the point \((h,k)\), where \(h = -\frac{b}{2a}\) and \(k = f(h)\). As b is zero, \(h = -\frac{0}{2*1/2} = 0\). Now to get k, substitute the h value in the quadratic equation: \(k = f(0) = \frac{1}{2} *(0)^{2}-5 = -5\). So the vertex of the parabola is at point (0, -5).

Step 3: Describe the graph of the quadratic function

The graph of the function \(f(x)=\frac{1}{2} x^{2}-5\) is a parabola. Since the coefficient of \(x^{2}\) is positive, the parabola opens upwards. This means it has a minimum value, located at the vertex. Hence, the graph of the function opens up and the vertex is the minimum point.

Step 4: Verify results with a graphing utility

Rely on a graphing utility to plot function \(f(x)=\frac{1}{2} x^{2}-5\). Confirm that the vertex is (0,-5) and the parabola opens upwards.

Key Concepts

quadratic function

A quadratic function is a specific type of polynomial function that takes the general form of \[ f(x) = ax^2 + bx + c \] where:

• \(a\), \(b\), and \(c\) are constants,
• \(a\) is not equal to zero.

This function represents a parabola when graphed on the Cartesian plane.
The orientation and shape of the parabola are determined by the coefficient \(a\).
If \(a\) is positive, the parabola opens upward, and if \(a\) is negative, it opens downward.
In our example, the quadratic function is given as \(f(x)=\frac{1}{2} x^{2}-5\), which can be classified in this form with \(a=\frac{1}{2}\), \(b=0\), and \(c=-5\).

vertex of a parabola

The vertex of a parabola is a crucial feature, as it represents either the highest or lowest point on the graph depending on the parabola's direction.
It is found at the point \((h,k)\).

• To find \(h\), use the formula \(h = -\frac{b}{2a}\).
• To find \(k\), substitute \(h\) into the function \(f(x)\) to get \(k = f(h)\).

For our specific function \(f(x)=\frac{1}{2}x^{2}-5\), since \(b=0\), \(h\) is calculated as zero: \[ h = -\frac{0}{2 \cdot \frac{1}{2}} = 0 \] To find \(k\), plug \(h=0\) into the function: \[ k = f(0) = \frac{1}{2}(0)^2 - 5 = -5 \] Hence, the vertex is at \((0, -5)\), which is the parabola's lowest point since it opens upwards.

graphing utility

A graphing utility is a versatile and powerful tool used to visually represent mathematical functions.
It allows you to plot functions like the quadratic function we're dealing with and observe their behaviors, such as intercepts and the vertex.
Using a graphing utility, you can:

• Input the function to see its graph plotted instantly.
• Identify the vertex and confirm its coordinates.
• Verify the direction in which the parabola opens.

By plotting the example function \(f(x)=\frac{1}{2} x^{2}-5\), a graphing utility can confirm that the vertex is indeed at the point \((0,-5)\) and that the parabola opens upwards.

upward-opening parabola

An upward-opening parabola is characterized by the positive coefficient of its \(x^2\) term, indicating that it curves upwards on the graph.
For the function \(f(x)=\frac{1}{2}x^2-5\), since \(a = \frac{1}{2}\) (a positive value), the parabola opens upwards.

• This means it has a minimum point at its vertex.
• The graph begins at the vertex moving symmetrically upward on both sides.
• Its arms extend towards positive infinity as \(x\) moves away from the vertex in either direction.

Understanding this concept is helpful, as it tells you about the function's behavior just by looking at the coefficient of \(x^2\).
Thus, for any quadratic function where \(a\) is greater than zero, the function will produce an upward-opening parabola.