Problem 54 - College Algebra

Question

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)=x^{2}-1 $$

Step-by-Step Solution

Answer

The function \(g(x)=x^{2}-1\) is a downward shift of the standard quadratic function \(f(x)=x^{2}\). To graph \(g(x)\), move every point on the \(f(x)\) graph down by one unit. This turns the vertex of the \(f(x)\) graph from (0,0) to (0,-1) in the \(g(x)\) graph.

Step 1: Understand the Standard Function

The standard quadratic function is given by \(f(x) = x^2\). This function produces a graph called a parabola, which is U-shaped. The graph of this function will pass through the origin (0, 0).

Step 2: Plot the Standard Function

Draw the \(f(x) = x^2\) parabola. Note that the points (1,1) and (-1,1) are also plotted, because these points will help us compare this graph to the graph of the transformed function.

Step 3: Understand the Transformation

The given function \(g(x) = x^2 - 1\) is a transformation of \(f(x) = x^2\). This involves subtracting 1 from each of the y-values of the \(f(x)\) graph. This is called a vertical shift. Since we subtract, it is a vertical shift downwards by 1 unit.

Step 4: Draw the Transformed Function

Perform the vertical shift by moving each point on the \(f(x)\) graph down by 1 unit. Plot these transformed points on the same set of axes. This transformed graph represents the function \(g(x) = x^2 - 1\).

Key Concepts

Standard Quadratic Function

A standard quadratic function serves as the foundational building block for understanding more complex quadratics. It is represented by the formula \(f(x) = x^2\). When graphed, this function produces a characteristic U-shaped curve known as a parabola. The most distinguishing feature of this parabola is its symmetry about the vertical axis and its vertex at the origin, which is the point (0,0).

The simple nature of the standard quadratic function makes it ideal for studying transformations, as other quadratic functions are often this standard equation with additional terms. By starting with the standard parabola, students can visually and algebraically discern the effects of different transformations.

Transformations of Graphs

Transformation of graphs is a concept that involves altering the appearance of a graph through various operations without changing the original function's input-output relationship. The primary transformations include translations (shifts), reflections, stretches, and compressions. In the context of quadratic functions, these transformations can move the graph up or down (vertical shift), left or right (horizontal shift), flip it across an axis (reflection), or alter its width (stretch or compression).

Understanding these transformations allows students to predict changes in the graph based on alterations to the function's equation. For instance, altering the square term's coefficient affects the parabola's width, while adding a constant term results in a vertical shift. Thus, recognizing the underlying standard function within a transformed quadratic is crucial for efficient graphing and analysis.

Vertical Shift

When graphing quadratic functions, one of the simplest transformations to identify and apply is the vertical shift. This movement occurs when a constant is added to or subtracted from the standard quadratic function, denoted as \(y = x^2 + k\) or \(y = x^2 - k\), where \(k\) represents the number of units the graph shifts vertically.

For the given exercise, the function to graph is \(g(x) = x^2 - 1\), which represents a vertical shift one unit downward from the standard parabola \(f(x) = x^2\). Such shifts maintain the shape and orientation of the graph, merely changing its position relative to the x-axis. Vertical shifts are often one of the first transformations students learn, as they provide a clear visual understanding of how equations translate into graphical changes.