Problem 21 - 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. \(f(x)=(x+3)^{2}-4\)

Step-by-Step Solution

Answer

The graph of the function \(f(x) = (x + 3)^2 - 4\) is a parabola opening upwards, with the vertex at point (-3, -4). A graphing utility can be used to verify these results.

Step 1: Identify the Form

The given function \(f(x) = (x + 3)^2 - 4\) is in the form \(f(x) = a(x - h)^2 + k\). Here, a = 1, h = -3 and k = -4.

Step 2: Find the Vertex

The vertex of function \(f(x) = a(x - h)^2 + k\) is given by point \(h, k\). In this case, the vertex will be \(-3, -4\).

Step 3: Describe the Graph

Since the value of a is positive, the function \(f(x) = (x + 3)^2 - 4\) will open upwards in a U-shape. This shape is called a parabola. The vertex is the lowest point on the graph, which is \(-3, -4\). The graph will be symmetric around the line \(x = -3\).

Step 4: Verify Using a Graphing Utility

Finally, use a graphing utility to plot the function \(f(x) = (x + 3)^2 - 4\). The plotted graph should support the analysis and match the description.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial when analyzing or graphing quadratic functions. The parabolic equation in the form \(f(x) = a(x - h)^2 + k\) reveals that the vertex of the parabola is the point \(h, k\). This is the highest or lowest point of the parabola, known as the maximum or minimum respectively, depending on the value of \(a\).

For the quadratic function \(f(x) = (x + 3)^2 - 4\), we identify the vertex by comparing it to the standard form. Recognizing that \(h = -3\) and \(k = -4\), the vertex of this parabola is at point \( (-3, -4) \). This specific vertex also determines the axis of symmetry and the direction in which the parabola opens.

When graphing a quadratic function, starting from the vertex allows for an easier sketch of the parabola, since you know where the curve turns and the direction in which it opens. In this case, the positive coefficient of the quadratic term \(a = 1\) ensures that the parabola opens upward.

Parabola Symmetry

Parabolas exhibit a beautiful characteristic known as symmetry. This means that if you draw a vertical line through the vertex, called the axis of symmetry, the parabola reflects perfectly across this line. The general equation for this axis is \(x = h\), where \(h\) is the x-coordinate of the parabola's vertex.

In the equation \(f(x) = (x + 3)^2 - 4\), with vertex \( (-3, -4) \), the axis of symmetry is the line \(x = -3\). The graph will show that every point on the left side of the axis will have a corresponding point with the same distance on the right side.

When graphing by hand, plotting points equally spaced from the axis of symmetry and connecting them with a smooth curve can create an accurate sketch of the parabola. Parabola symmetry also aids in solving real-life problems where reflecting trajectories or pathways need to be calculated, like in physics or engineering.

Graphing Utility

A graphing utility is a tool that aids in the visualization of mathematical functions and their properties. It can provide a precise and accurate graph based on the function's equation and is especially helpful for complex functions that are difficult to sketch by hand.

For the function \(f(x) = (x + 3)^2 - 4\), a graphing utility such as a calculator app or software can help verify your sketched graph. You would input the given function and observe how the graphing utility plots the curve, ideally confirming the vertex at \( (-3, -4) \), the upward opening, and the symmetry about the axis \(x = -3\).

Utilizing such tools not only confirms your understanding and calculations but can also enhance learning by visually demonstrating the impact of changing different parameters in the equation. It's an essential aid for students to validate their work and gain confidence in the concepts of graphing quadratic functions.