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Chapter 2: Problem 54

A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x| ;\) shrink vertically by a factor of \(\frac{1}{2},\) shift to the left 1 unit, and shift upward 3 units

Short Answer

Expert verified
The final function is \(k(x) = \frac{1}{2}|x+1| + 3\).

Step by step solution

01

Identify the Base Function

The base function given is \(f(x) = |x|\), which is the absolute value function.
02

Apply Vertical Shrink

The function is shrunk vertically by a factor of \(\frac{1}{2}\). To apply this transformation, multiply the function by \(\frac{1}{2}\). The new function becomes \(g(x) = \frac{1}{2}|x|\).
03

Apply Horizontal Shift

Shift the function to the left by 1 unit. To do this, replace \(x\) with \(x+1\) in the function. The transformed function becomes \(h(x) = \frac{1}{2}|x+1|\).
04

Apply Vertical Shift

Finally, shift the function upward by 3 units. To accomplish this, add 3 to the function. The final transformed function is \(k(x) = \frac{1}{2}|x+1| + 3\).

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Key Concepts

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

Absolute Value Function
The absolute value function is a cornerstone in mathematics and is written as \(f(x) = |x|\). This function creates a "V" shape on a graph. The graph has a vertex at the origin, which is the point \((0,0)\). It mirrors symmetrically around the y-axis.
Here are some key features of the absolute value function:
  • It reflects negative values of \(x\) to positive outputs, ensuring all results are non-negative.
  • The slope of each arm of the "V" is \(1\) when graphed.
  • The function is continuous and piecewise linear.
Understanding the basics of this function is critical when performing transformations such as shifts or shrinks in subsequent operations.
Vertical Shrink
A vertical shrink affects how tall or compressed a graph looks while keeping its width the same. It is performed by multiplying the function with a constant factor. In our exercise, we performed a vertical shrink by a factor of \(\frac{1}{2}\).
Here's what you do during a vertical shrink:
  • Multiply the entire function by the shrink factor, which in this case is \(\frac{1}{2}\).
  • Every y-coordinate on the graph is multiplied by this factor, making the graph appear half as tall.
  • This doesn't change the zeros or the x-intercepts of the function.
In the transformed function, this step took the original \(f(x) = |x| \) and changed it to \(g(x) = \frac{1}{2}|x|\). It's a simple yet impactful change to how the graph looks.
Horizontal Shift
A horizontal shift moves the entire graph left or right on the x-axis. This transformation is achieved by adding to or subtracting from the \(x\) variable within the function. For our exercise, we shifted the graph to the left by 1 unit.
Key points about horizontal shifts:
  • For a shift left, add the number of units to \(x\) (e.g., \(x+1\)).
  • For a shift right, subtract the number of units from \(x\) (e.g., \(x-1\)).
  • This step operates on the variable directly, influencing the input values of the function.
When applied to our shrunk function, it transformed \(g(x) = \frac{1}{2}|x|\) into \(h(x) = \frac{1}{2}|x+1|\), shifting the "V" to start one unit left of its original position.
Vertical Shift
The vertical shift moves the graph upward or downward along the y-axis without affecting its shape. This is done by adding or subtracting a constant at the end of the function. In our case, we shifted the function upward by 3 units.
Steps and features of a vertical shift:
  • Add a constant to move up or subtract to move down.
  • This impacts the entire graph equally, elevating or lowering the "V".
  • The overall shape remains unchanged, but the vertex's position changes.
Applying this last adjustment to our function resulted in \(k(x) = \frac{1}{2}|x+1| + 3\). Each point on the graph, including the vertex, moved up three units along the y-axis, completing the transformation process.

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Most popular questions from this chapter

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