Problem 18 - College Algebra

Question

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 2 units right and 3 units up.

Step 1: Understand the Original Function

Start with the basic function graph of \( y=f(x) \). This is the original function that we will be transforming.

Step 2: Horizontal Shift

The function \( y=f(x-2) \) indicates a horizontal shift to the right. The graph of \( y=f(x) \) is shifted 2 units to the right. This occurs because the input \( x \) is replaced by \( x-2 \), implying that each point moves 2 units along the positive \( x \)-axis.

Step 3: Vertical Shift

After applying the horizontal shift, the function \( y=f(x-2) \) is further transformed by adding 3, resulting in \( y=f(x-2)+3 \). This operation shifts the graph vertically upward by 3 units. Adding a constant outside the function represents a movement along the \( y \)-axis.

Step 4: Combine Transformations

The final graph \( y=f(x-2)+3 \) is obtained by applying both transformations. First, shift the graph of \( y=f(x) \) 2 units to the right, and then shift it 3 units upwards. The sequence of transformations affects the final position of the graph.

Key Concepts

Horizontal Shift

In graph transformations, a horizontal shift changes the position of a graph along the x-axis. When we have a function such as \(y = f(x - 2)\), it involves replacing \(x\) in the original function \(y = f(x)\) with \(x - 2\). This means every point on the graph of \(y = f(x)\) is moved 2 units to the right on the x-axis. This shift happens because you're telling the graph to "wait" two more units before taking action—essentially delaying all x-values.

• The shift is to the right when we subtract a positive constant from \(x\), like \(x - 2\).
• If we added a constant instead (like \(x + 2\)), the shift would be to the left.

This might seem counterintuitive because subtracting moves the graph to the right, but it all has to do with the inputs of the function adjusting to accommodate the new \(x\) values.

Vertical Shift

Just like the horizontal shift moves the graph left or right along the x-axis, a vertical shift moves the graph up or down along the y-axis. In the case of \(y = f(x - 2) + 3\), the \(+3\) outside the function indicates a vertical shift upwards.

• When you add a number to the function, the graph shifts upward by that number of units.
• Conversely, subtracting a number would move the graph downward.

This transformation affect all points on the graph equally, raising or lowering their y-values by the same increment.
So in our example, each y-value from the function \(y = f(x - 2)\) is increased by 3, indicating a movement 3 units up.

Function Graphs

Function graphs are visual representations of mathematical functions, allowing us to see how the outputs (or y-values) change as the inputs (x-values) vary. These graphs can represent all kinds of functions, including linear, quadratic, exponential, and more.

• Studying function graphs helps to visualize transformations, such as shifts, stretches, and reflections.
• Understanding a function's graph is key in seeing how different transformations affect it visually.

For example, the function \(y = f(x - 2) + 3\), starts with its base graph \(y = f(x)\). Transformations change certain characteristics, such as its location, but not its general shape. By examining these transformations in graph form, students can better grasp how functions behave.

Transformation Sequence

Transformation sequences involve applying multiple changes to a function graph, one after another. The order of these transformations can significantly influence the result. For instance, with \(y = f(x - 2) + 3\), the sequence involves:

• First, applying a horizontal shift 2 units to the right: this modifies the input value \(x\) to \(x - 2\).
• Second, applying a vertical shift upward by 3 units: this adds 3 to each y-value of the horizontally shifted function.

Through this specific sequence, the graph ends up in a new position on the plane without altering its overall shape. Understanding this concept helps to anticipate and understand the effect of compound transformations on graphs.
By practicing transformation sequences, students can achieve a deeper understanding of how various operations influence function behavior.