Problem 30 - Elementary and Intermediate Algebra

Question

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples \(2,3,\) and 4. $$ f(x)=x^{2}-4 $$

Step-by-Step Solution

Answer

The domain is \((-\infty, \infty)\) and the range is \([-4, \infty)\).

Step 1: Create a Table of Values

To graph the function, start by creating a table with several values of \(x\) and their corresponding \(f(x)\) values. Select a few integers around zero, such as \(-3, -2, -1, 0, 1, 2,\) and \(3\). Compute \(f(x) = x^2 - 4\) for each selected \(x\).

Step 2: Calculate Function Values

Use the function to calculate each \(f(x)\): - For \(x = -3\), \(f(-3) = (-3)^2 - 4 = 9 - 4 = 5\). - For \(x = -2\), \(f(-2) = (-2)^2 - 4 = 4 - 4 = 0\). - For \(x = -1\), \(f(-1) = (-1)^2 - 4 = 1 - 4 = -3\). - For \(x = 0\), \(f(0) = 0^2 - 4 = -4\). - For \(x = 1\), \(f(1) = 1^2 - 4 = -3\). - For \(x = 2\), \(f(2) = 2^2 - 4 = 4 - 4 = 0\). - For \(x = 3\), \(f(3) = 3^2 - 4 = 9 - 4 = 5\).

Step 3: Plot the Points

Plot the points \((-3, 5)\), \((-2, 0)\), \((-1, -3)\), \((0, -4)\), \((1, -3)\), \((2, 0)\), and \((3, 5)\) on a coordinate plane. These points will form the parabola shape of the quadratic function.

Step 4: Draw the Graph

Draw a smooth curve through the plotted points to form the parabola. The graph is symmetrical about the y-axis and opens upwards, with its vertex at \((0, -4)\).

Step 5: Determine the Domain and Range

The domain of the function \(f(x) = x^2 - 4\) is all real numbers, written as \((-\infty, \infty)\). The range is given by the set of all possible \(f(x)\) values, which starts from the vertex at \(-4\) and extends to positive infinity. Thus, the range is \([-4, \infty)\).

Key Concepts

Graphing Parabolas

When graphing parabolas, the main challenge is to represent the quadratic function visually. A parabola is a symmetrical curve that opens upward or downward. In this specific case of the function \( f(x) = x^2 - 4 \), the parabola opens upwards because the coefficient of \( x^2 \) is positive. To graph a parabola:

• Create a table of values, selecting some x-values, compute the corresponding y-values \( f(x) \), and plot these points on a coordinate plane.
• Once the points are plotted, draw a smooth curve through them. This is your parabola.

It’s important to note that the parabola is symmetrical about its vertex. This symmetry helps in predicting the shape of the graph.
For \( f(x) = x^2 - 4 \), we're dealing with a standard parabola that’s been shifted vertically downward by 4 units.

Domain and Range

In mathematics, the term domain refers to all possible input values (x-values) that a function can accept, while the range refers to all possible output values (f(x) or y-values) that a function can produce after being evaluated. For the function \( f(x) = x^2 - 4 \):

• The domain is all real numbers \,\((-\infty, \infty)\),\ because you can substitute any real number into \( x^2 - 4 \).
• The range is limited to \([-4, \infty)\),\ because the lowest value the function can reach is at the vertex, \(-4\), and it can go as high as infinity.

Understanding domain and range helps in analyzing the extent and behavior of the function. A common mistake is confusing domain with range, so always remember: domain refers to x-values, range refers to y-values.

Coordinate Plane

A coordinate plane is a two-dimensional surface where each point is determined by an ordered pair of numbers \((x, y)\). In graphing, having a solid understanding of the coordinate plane is crucial. It consists of:

• The horizontal axis, called the x-axis.
• The vertical axis, called the y-axis.
• The point where they intersect is the origin \((0, 0)\).

The coordinate plane allows you to systematically plot points and observe the shape and features of functions. For the function \( f(x) = x^2 - 4 \), you'll plot points such as \((-3, 5)\) and \((0, -4)\). These points align to show the shape and location of the parabola on the plane.

Parabola Vertex

The vertex of a parabola is a pivotal feature, representing the peak or lowest point on the graph. For the parabola derived from \( f(x) = x^2 - 4 \), the vertex is at \((0, -4)\). Here’s why it's significant:

• It is the point of symmetry for the parabola, meaning the curve is mirrored each side of this point.
• In this graph, the vertex is the lowest point, as the parabola opens upwards.
• To find the vertex of \(Ax^2+Bx+C=0\), the formula is \(x = \frac{-B}{2A}\), but here B is 0, so it's straightforward \(x=0\).

The vertex gives insight into the graph's shape and can help in identifying the minimum or maximum values of the function. Understanding the position of the vertex is important for graphing accurately.