Problem 85 - 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 \(y=g(x)\). \(g(x)=f(x-2)+1\)

Step-by-Step Solution

Answer

The points are (-10, 7), (2, 9), and (10, -3).

Step 1: Understand the Transformation

The function transformation applied to obtain \( g(x) \) from \( f(x) \) is \( g(x) = f(x-2) + 1 \). This indicates a horizontal shift to the right by 2 units and an upward shift by 1 unit.

Step 2: Transform the First Point

Start with the point \((-12, 6)\). To find the corresponding point on \(g(x)\), shift the x-coordinate 2 units to the right to get \(-10\), and shift the y-coordinate up by 1 unit to get \(7\). So, the point becomes \((-10, 7)\).

Step 3: Transform the Second Point

Take the second point \((0, 8)\) and apply the transformation. The x-coordinate becomes \(0+2=2\), and the y-coordinate becomes \(8+1=9\). Thus, the point is \((2, 9)\).

Step 4: Transform the Third Point

For the third point \((8, -4)\), shift the x-coordinate to \(8+2=10\), and the y-coordinate to \(-4+1=-3\). This yields the point \((10, -3)\) on \(g(x)\).

Key Concepts

Graph Transformations

When we discuss graph transformations, we talk about changes that can be made to the graph of a function. These transformations could shift, stretch, compress, or reflect the graph.
A common transformation involves shifting, which is simply moving the graph to a different position in the coordinate plane. This is visually altering the appearance but not the shape of the graph.
Transformations can be:

• Horizontal shifts: Move the graph left or right.
• Vertical shifts: Move the graph up or down.
• Reflections: Flip the graph over an axis.
• Stretches and compressions: Alter the size of the graph.

Understanding these movements allows us to predict how the graph will look without having to plot it point by point.

Algebraic Functions

Algebraic functions are expressions made up of variables and constants using the operations of addition, subtraction, multiplication, division, and exponentiation with whole number exponents.
They can be viewed as a rule that relates an input to a specific output.
In this context, the function \( y = f(x) \) represents an algebraic function from which we derive \( y = g(x) \) through transformation.

• Algebraic functions like linear, quadratic, or polynomial functions often form the basis of graphs.
• By modifying these functions using transformations, we can create new graphs, such as in the shift from \( f(x) \) to \( g(x) \).

This manipulation allows us to explore different dynamic relationships between variables in a visual format.

Coping with Shifts in Functions

To cope with shifts in functions, it's crucial to understand how each transformation impacts the graph. When you see an equation like \( g(x) = f(x-2) + 1 \), you're dealing with specific shifts:

• Horizontal Shift: The expression \( (x-2) \) indicates that for every point, the x-coordinate is moved two units to the right.
• Vertical Shift: Adding 1 to the function \( f(x) \) means every y-coordinate goes up by 1 unit.

These shifts affect where the points lie on the graph but maintain the general shape and structure of the original function, \( f(x) \).
Coping with these shifts involves recalculating the coordinates of key points to predict where they will appear on the new graph. This understanding aids in visualizing how transformations will affect real-world situations or data.