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Chapter 3: Problem 31

Find the function that is finally graphed after each of the following transformations is applied to the graph of \(y=\sqrt{x}\) in the order stated. (1) Vertical stretch by a factor of 3 (2) Shift up 4 units (3) Shift left 5 units

Short Answer

Expert verified
The final function is \(y = 3\sqrt{x + 5} + 4\).

Step by step solution

01

Apply the Vertical Stretch

Start with the base function: \(y = \sqrt{x}\). Apply the vertical stretch by a factor of 3: \(y = 3\sqrt{x}\). This multiples the output of the square root function by 3.
02

Apply the Vertical Shift

Take the function obtained from Step 1: \(y = 3\sqrt{x}\). Shift this function up by 4 units: \(y = 3\sqrt{x} + 4\). This adds 4 to the output of the vertically stretched function.
03

Apply the Horizontal Shift

Take the function obtained from Step 2: \(y = 3\sqrt{x} + 4\). Shift this function left by 5 units. Subtract 5 from the input of the function: \(y = 3\sqrt{x + 5} + 4\). This moves the graph 5 units to the left.

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Key Concepts

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

Vertical Stretch
A vertical stretch involves stretching the graph of a function away from the x-axis. It changes the shape but not the position of the graph. In the context of our example, we start with the base function: \(y = \sqrt{x}\). To apply a vertical stretch by a factor of 3, we multiply the entire output of the function by 3: \(y = 3\sqrt{x}\). This means every y-value of the base function \(\sqrt{x}\) is now three times higher.
  • Imagine pulling the graph upwards by a factor of 3 along the y-axis while keeping the x-values unchanged.
  • This operation emphasizes the effect of the transformation on the graph鈥檚 steepness.
By understanding the concept of vertical stretch, you can easily modify various graphs to better fit the required transformations.
Vertical Shift
A vertical shift moves the graph of a function up or down along the y-axis without changing its shape. For example, if we take the stretched function \(y = 3\sqrt{x}\) and shift it up by 4 units, we get \(y = 3\sqrt{x} + 4\). This transformation involves adding a constant to the function鈥檚 output:
  • Every point on the graph of \(y = 3\sqrt{x}\) is moved up by 4 units.
  • The overall shape and steepness of the graph remain unchanged.
  • The new graph \(y = 3\sqrt{x} + 4\) simply sits 4 units higher than the original graph.
Vertical shifts are useful for adjusting the position of the graph relative to the y-axis without affecting its general appearance.
Horizontal Shift
A horizontal shift involves moving the graph left or right along the x-axis. It alters the position but not the shape of the graph. Using our example, we take the vertically shifted function \(y = 3\sqrt{x} + 4\) and shift it left by 5 units. This is done by replacing \(x\) with \(x + 5\) in the function, resulting in \(y = 3\sqrt{x + 5} + 4\). Here鈥檚 what happens:
  • The input \(x\) is adjusted by adding 5, causing the graph to move left.
  • In effect, every point on the graph is translated 5 units to the left.
  • The shape and vertical position of the graph are preserved.
Understanding horizontal shifts helps you reposition graphs along the x-axis for different functional transformations.
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to visualize the behavior of the function. Each transformation applied reflects a different modification to this graph. Let鈥檚 consider the final function \(y = 3\sqrt{x + 5} + 4\):
  • Start with the base graph of \(y = \sqrt{x}\).
  • Apply the vertical stretch to get: \(y = 3\sqrt{x}\).
  • Shift the graph up by 4 units resulting in: \(y = 3\sqrt{x} + 4\).
  • Finally, shift the graph left by 5 units, resulting in: \(y = 3\sqrt{x + 5} + 4\).
By understanding and applying each transformation step-by-step, it becomes easier to accurately graph even the most complex functions. Visualizing these changes clearly shows how each transformation affects the position and shape of the graph.

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