Problem 3 - Intermediate Algebra

Question

Graph each of the following linear and quadratic functions. $$f(x)=-2 x^{2}$$

Step-by-Step Solution

Answer

The graph is a downward-opening parabola with vertex at (0, 0).

Step 1: Understand the Function Type

The given function is a quadratic function of the form \(f(x) = ax^2 + bx + c\). In this case, \(a = -2\), \(b = 0\), and \(c = 0\). This indicates the graph will be a parabola that opens downward because the coefficient \(a\) is negative.

Step 2: Determine the Vertex

In a quadratic function \(f(x) = ax^2 + bx + c\), the vertex can be found using the formula \(x = -\frac{b}{2a}\). For this function, since \(b = 0\), the vertex of the parabola is at \(x = 0\). Substitute \(x = 0\) into \(f(x)\) to find \(f(0) = -2(0)^2 = 0\). Thus, the vertex is at the point (0, 0).

Step 3: Find Additional Points

To better sketch the parabola, calculate additional points on either side of the vertex. Choose \(x = 1\) and \(x = -1\) for simplicity: \(f(1) = -2(1)^2 = -2\), so the point is (1, -2). \(f(-1) = -2(-1)^2 = -2\), giving the point (-1, -2).

Step 4: Plot the Vertex and Points

Plot the vertex (0, 0) and the additional points (1, -2) and (-1, -2) on a coordinate plane. These points assist in shaping the parabola.

Step 5: Draw the Parabola

Using the plotted points, draw a smooth curve through these points. The parabola should open downward and be symmetric around the y-axis due to the nature of the quadratic function \(f(x) = -2x^2\). Include arrows on either end of the parabola to indicate it continues indefinitely.

Key Concepts

Understanding a Parabola

A parabola is a unique, U-shaped curve that can open either upward or downward, depending on the function of its equation. In this case, we have a quadratic function which is written in the form \(f(x) = ax^2 + bx + c\). The graph of any quadratic function is a parabola. The direction in which the parabola opens is determined by the sign of the coefficient \(a\).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

For our specific function \(f(x) = -2x^2\), \(a = -2\), so the parabola opens downward. This downward-opening characteristic makes the vertex of the parabola the highest point, also known as the maximum point.

Finding the Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction. It is a key feature of quadratic graphs.
The vertex can be easily identified using the formula \(x = -\frac{b}{2a}\) in the general quadratic equation \(f(x) = ax^2 + bx + c\). For the function we are working with, \(b = 0\), which simplifies finding the vertex:

• Using the formula, \(x = -\frac{0}{2(-2)} = 0\)
• Substituting \(x = 0\) back into the original equation gives \(f(0) = -2(0)^2 = 0\)
• This means our vertex is at (0, 0)

The vertex provides a reference point to shape the parabola, especially crucial since it helps in plotting the initial direction and curve of the graph.

Plotting Points for a Parabola

Plotting points is essential for graphing the precise shape of the parabola. Starting with the vertex is useful, but additional points provide clarity on the curve’s path.
For our function \(f(x) = -2x^2\), it’s best to choose values around the vertex to understand how the parabola behaves:

• Start with \(x = 1\). Calculating gives \(f(1) = -2(1)^2 = -2\). Plot point (1, -2).
• Next try \(x = -1\). Here, \(f(-1) = -2(-1)^2 = -2\). Plot point (-1, -2).

These additional points reflect the symmetry of the parabola around the y-axis. Plotting these helps demonstrate the curve smoothly and symmetrically from the vertex outward.
Once familiar with plotting these points, draw a smooth curve through them. Always note the endpoints extend indefinitely, showcasing a parabola's continuous path.