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Chapter 1: Problem 99

How do the graphs of \(f(x)\) and \(f(x+10)\) differ?

Short Answer

Expert verified
The graph of \(f(x+10)\) is the graph of \(f(x)\) shifted 10 units left.

Step by step solution

01

Understanding Function Translation

The function translation affects the position of the graph without changing its shape. When a value is added or subtracted inside the function argument, it translates the graph horizontally.
02

Identifying Horizontal Translation

For the function \(f(x+10)\), the \(+10\) inside the argument of the function indicates a horizontal translation. Specifically, it shifts the graph of \(f(x)\) to the left by 10 units.
03

Graph Comparison

Draw the graph of \(f(x)\), then shift the entire graph 10 units to the left to get the graph of \(f(x+10)\). The shape remains unchanged, and only the position along the x-axis is altered.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Translation
When you see a function like \( f(x+10) \), it's a clue that a horizontal translation is at play. This means the entire graph of the function \( f(x) \) is going to be shifted along the x-axis. The direction of the shift depends on the sign of the number inside the function argument. Here, the presence of \(+10\) actually moves the graph 10 units to the left.

It might seem counterintuitive that \(+10\) means moving left, but remember, adding a positive number inside the function argument \( f(x) \) makes all x-values in the function effectively start earlier, hence moving the graph leftward.
Graph Transformation
Graph transformations involve altering the position, size, or orientation of a graph in various ways. In the context of this exercise, we focus on horizontal shifts which are a type of transformation.

Unlike other transformations such as scaling or reflection, horizontal translation affects only the graph's position without changing its size or orientation.
  • A positive shift (\( f(x-c) \)) inside the function moves the graph c units to the right.
  • A negative shift (\( f(x+c) \)) moves it c units to the left.

These transformations are useful for adjusting graphs to fit specific contexts or to compare different functions.
Function Shifts
Function shifts include both horizontal and vertical adjustments. Horizontal shifts, as explained earlier, involve moving the function along the x-axis. But let's delve a bit deeper into how these shifts occur and what they imply.

The input values of the function鈥攂ut not the outputs鈥攁re directly affected by the horizontal shifts. This means the graph slides left or right, altering where the function appears along the x-axis. Conversely, vertical shifts would modify the output values, moving the graph up or down along the y-axis.
  • Horizontal shifts: Change in position along the x-axis without effect on y-values.
  • Vertical shifts: Adjustments in y-values while x-values are kept constant.

Understanding these shifts helps you quickly sketch or analyze functions without recalculating everything from scratch.

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