Problem 45 - College Algebra with Modeling and Visualization

Question

Use transformations to sketch a graph of \(f\). \(f(x)=x^{2}-3\)

Step-by-Step Solution

Answer

The graph of \(f(x) = x^2 - 3\) is a parabola opening upwards with vertex at \((0, -3)\).

Step 1: Identify the Basic Function

The function given is \(f(x) = x^2 - 3\). The basic function we start with is \(y = x^2\), which is a standard quadratic function and forms a parabola opening upwards with its vertex at the origin \((0, 0)\).

Step 2: Determine the Transformation

The function \(f(x) = x^2 - 3\) indicates a vertical transformation. Specifically, \(-3\) represents a vertical shift. This means we shift the parabola \(3\) units down along the y-axis.

Step 3: Apply the Transformation

To apply the transformation, take each point on the graph of \(y = x^2\) and move it down by 3 units. Thus, the vertex of the parabola \(y = x^2\), originally at \((0, 0)\), will move to \((0, -3)\). Likewise, any other point \((x, y)\) on the original parabola will move to \((x, y-3)\).

Step 4: Sketch the Transformed Graph

Start by plotting the new vertex at \((0, -3)\). The parabola is symmetric about the y-axis. Points on the original graph such as \((1, 1)\) will move to \((1, -2)\) and \((-1, 1)\) will move to \((-1, -2)\). Draw a smooth curve through these points to sketch the new transformed parabola.

Key Concepts

Vertical Shift

A vertical shift in a graph occurs when the entire graph moves up or down along the y-axis. This kind of transformation is very straightforward; you simply adjust each y-coordinate by the amount indicated. In our example of function transformation for the equation \(f(x) = x^2 - 3\), the graph undergoes a vertical shift.

Here, the "-3" in the equation tells us to shift downward by 3 units. Such a shift means that every point \((x, y)\) on the graph of the basic \(y = x^2\) will now be at \((x, y-3)\). Think of this vertical shift as giving your whole graph a gentle nudge downward.

• The vertex of the original parabola at the origin \((0, 0)\) moves down to \((0, -3)\).
• This transformation does not affect the x-values, only the y-values.

Shifting like this can change the appearance of the graph and is important for correctly interpreting the function's behavior.

Parabola Graphing

Graphing a parabola involves recognizing its fundamental shape. A parabola is usually associated with quadratic functions of the form \(y = ax^2 + bx + c\). These functions produce a "U" shaped curve on the graph known as a parabola.

For the simplest case, the parabola \(y = x^2\), is perfectly symmetrical around the y-axis with its vertex at the origin. The parameter "a" in the general quadratic equation affects the width and direction of the opening:

• When \(a > 0\), the parabola opens upward.
• When \(a < 0\), it opens downward.

When graphing, it is useful to:

• Identify the vertex as it acts like the "center" of the parabola.
• Recognize symmetry, which allows you to predict points on one side based on points from the other.

In the context of \(f(x) = x^2 - 3\), you're simply shifting the entire curve vertically without altering its shape or direction.

Graph Transformations

Graph transformations allow for manipulating a basic graph to match more complex equations. These transformations can involve translations, reflections, stretching, or compressing. The function \(f(x) = x^2 - 3\) is an example of a vertical translation, a type of graph transformation.

Understanding graph transformations begins with identifying the type of transformation:

• Vertical shifts, such as \(-3\) in \(x^2 - 3\), move the graph up or down.
• Horizontal shifts would move the graph left or right, but these appear as transformations in the formula \(x - h\).
• Reflections can flip a graph over the x or y-axis and are indicated by negative signs.
• Stretching or compressing affect the width of the graph and are usually indicated by multiplying \(x\) or \(y\) by a constant.

Graph transformations are powerful tools in graphing functions as they demonstrate changes in behavior and positioning without altering the intrinsic properties of the original graph.