Problem 83 - College Algebra with Modeling and Visualization

Question

The points \((-12,6),(0,8),\) and \((8,-4)\) lie on the graph of \(y=f(x)\). Determine three points that lie on the graph of \(y=g(x)\). \(g(x)=f(x)+2\)

Step-by-Step Solution

Answer

Points: (-12, 8), (0, 10), (8, -2)

Step 1: Identifying Given Points

Identify the coordinates of points given on the graph of \(y = f(x)\). These points are \((-12, 6)\), \((0, 8)\), and \((8, -4)\).

Step 2: Understanding the Transformation

Understand that \(g(x) = f(x) + 2\) means that to find \(y\) values on the graph of \(g(x)\), you add 2 to each corresponding \(y\) value from the graph of \(f(x)\).

Step 3: Applying the Transformation to Each Point

1. Starting with \((-12, 6)\), the new \(y\) value will be \(6 + 2 = 8\). Thus, the corresponding point on \(y = g(x)\) is \((-12, 8)\).2. For \((0, 8)\), the new \(y\) value will be \(8 + 2 = 10\), giving the point \((0, 10)\).3. For \((8, -4)\), the new \(y\) value will be \(-4 + 2 = -2\), resulting in the point \((8, -2)\).

Key Concepts

Translation of Graph

Graph translations involve shifting the entire graph of a function in one direction without changing its shape. A key type of transformation is the vertical shift, which is directly related to this exercise.
This alteration is expressed in the functional form as adding or subtracting a constant to the function. For example, in our problem, the function transformation is given by:

• If you have a function \(f(x)\), and you define a new function \(g(x) = f(x) + 2\), this results in a vertical translation.
• Every point on the graph of \(f(x)\) is shifted up by 2 units to form the graph of \(g(x)\).

The beauty of this transformation lies in its simplicity: it only affects the \(y\)-values of the points, leaving the \(x\)-values unchanged.

Coordinate Points

Coordinate points form the basis of understanding graphs and are essential in plotting. A coordinate point \((x,y)\) represents a location on the Cartesian plane. The \(x\) value, or abscissa, refers to the horizontal position, while the \(y\) value, or ordinate, indicates the vertical position.

• In our exercise, we start with the given points:
  • \((-12, 6)\),
  • \((0, 8)\),
  • \((8, -4)\)

When we perform the vertical translation, only the \(y\)-values change. To compute the new points for the translated graph \(g(x) = f(x) + 2\):

• \((-12, 6)\) becomes \((-12, 8)\)
• \((0, 8)\) becomes \((0, 10)\)
• \((8, -4)\) becomes \((8, -2)\)

These simple calculations are a direct reflection of understanding shifts in coordinate points, demonstrating how each point on a graph shifts based on the function's transformation method.

Graph of a Function

A graph of a function visually represents all the set points \((x, y)\) satisfying the function equation. It provides a mechanism to see the detailed behavior of a function across different values. For a clear example:

• In the case of \(y = f(x)\), we initially plot the points such as \((-12, 6)\), representing the behavior of the original function.

As transformations like \(y = g(x)\), such as \(g(x) = f(x) + 2\), are applied, the graph changes accordingly. These adjustments enable new graphs to visualize shifts in function behavior.

• After translation, new points, like \((-12, 8)\), are plotted, detailing the function's modified state.

This graphical approach helps students and teachers alike to intuitively understand and analyze how functions transform, offering a tangible way to perceive function relationships through shifted graphs.