Problem 18 - College Algebra

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). $$ y=f(x-2)+3 $$

Step-by-Step Solution

Answer

The graph shifts to the right by 2 units and upward by 3 units.

Step 1: Identify the Original Function

The original function is given as \( f(x) \). This is the baseline function graph that we will be transforming.

Step 2: Identify the Horizontal Shift

The function \( f(x-2) \) indicates a horizontal shift. Since the expression is \( x-2 \), this shifts the original graph to the right by 2 units. In general, \( f(x-h) \) shifts the graph of \( f(x) \) to the right by \( h \) units if \( h \) is positive.

Step 3: Identify the Vertical Shift

The graph is also modified by adding 3 to the function: \( f(x-2) + 3 \). This addition results in a vertical shift upward by 3 units. In general, adding \( k \) to \( f(x) \) shifts the graphs of the function upward by \( k \) units if \( k \) is positive.

Step 4: Describe the Total Transformation

Combining both transformations, the original graph of \( f(x) \) is shifted to the right by 2 units and then shifted upward by 3 units.

Key Concepts

Horizontal Shift

A horizontal shift is when a graph moves left or right from its original position. In the expression \( y = f(x-2)+3 \), the horizontal shift is caused by \( f(x-2) \).
This notation means the graph of the original function \( f(x) \) shifts to the right by 2 units. To understand why study the point where \( x \) changes.

• If you replace \( x \) with \( x-2 \), the peak or valley of the graph (or any specific point on it) moves from \( x \) to \( x+2 \).
• This is because you need a larger \( x \) value to achieve the same \( f(x) \) height compared to the original graph.

Remember, if you see \( f(x-h) \), it's a right shift by \( h \) units; if it's \( f(x+h) \), it's a left shift by \( h \) units.

Vertical Shift

A vertical shift moves the graph up or down. This shift does not alter the shape of the graph; it simply raises or lowers it.
In our example \( y = f(x-2) + 3 \), the \(+3\) indicates a vertical shift of the original graph up by 3 units.

• The addition of 3 means every point on the graph moves 3 units higher.
• If it was \( -3 \), the graph would move 3 units downward.

Vertical shifts are straightforward—look for changes added or subtracted to \( f(x) \) to determine direction and magnitude.

Graph Translations

Graph translations encompass both horizontal and vertical shifts. When a function is translated, its shape remains unchanged, but its position alters.
In the expression \( y = f(x-2) + 3 \), the graph undergoes a translation.

• The horizontal shift moves it right by 2 units.
• The vertical shift lifts it up by 3 units.

To visualize, imagine picking up the graph and sliding it over and up.These translations are powerful because they help modify a graph's position without affecting how steep or wide the function may appear.

Original Function

The original function \( f(x) \) is like the base or starting point. Any transformation we discuss will be applied to this function.
In our example, \( f(x) \) could be any known function, such as \( x^2 \), \( \sin(x) \), or even a complex polynomial.

• Understanding the behavior of \( f(x) \) is important before translating.
• Transformations like shifts allow us to maneuver \( f(x) \) into different axes while retaining its identity.

Always identify the "unchanged" \( f(x) \) before embarking on shifts and transformations to make process easier.