Problem 53 - Beginning and Intermediate Algebra

Question

If the following transformations are performed on the graph of \(f(x)\) to obtain the graph of \(g(x),\) write the equation of \(g(x)\). \(f(x)=|x|\) is shifted left 2 units and down 1 unit.

Step-by-Step Solution

Answer

The equation of \(g(x)\) after shifting the graph of \(f(x) = |x|\) left 2 units and down 1 unit is \(g(x) = |x + 2| - 1\).

Step 1: Modify the input variable

To shift the graph of a function left by 2 units, we need to replace the input variable \(x\) with \((x+2)\). In our case, we replace \(x\) with \((x + 2)\) in the function \(f(x) = |x|\) giving us the new function: \(f(x) = |x+2|\) #Step 2: Shift the graph down 1 unit#

Step 2: Modify the output variable

To shift the graph of a function down by 1 unit, we need to subtract 1 from the entire function. In our case, we subtract 1 from the function we obtained in step 1: \(g(x) = |x + 2| - 1\) So, the equation of \(g(x)\) after applying both transformations is: \(g(x) = |x + 2| - 1\)

Key Concepts

Absolute Value Function

The absolute value function is a special type of function in mathematics. It is written as \(f(x) = |x|\), and its graph is shaped like a "V". This function takes any real number \(x\), and returns its non-negative version. For example, if \(x\) is -3, \(|x|\) would be 3.
This operation makes the graph symmetrical with respect to the y-axis. The vertex or the "point" of the V is always at the origin (0,0) unless transformations like shifts are applied. It's crucial to understand this function, as many transformations often start with the absolute value graph.

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right by a number of units. For the function \(f(x) = |x|\), if you want to move the graph left by 2 units, you replace \(x\) with \((x+2)\).
Why do we add, you might ask? Adding inside the absolute value shifts the graph in the opposite direction of most people’s first assumption.

• Adding a positive number shifts the graph left.
• Subtracting would shift it to the right.

The new function becomes \(f(x) = |x+2|\). This operation essentially modifies the inputs before they hit the absolute value operation, shifting the entire function in the horizontal direction.

Vertical Shift

Vertical shifts move the graph of a function up or down. For any function \(f(x)\), a vertical shift is performed by adding or subtracting a number outside of the function's operation.
For the function \(f(x) = |x+2|\) that we've horizontally shifted, if we want to move the graph down by 1 unit, we simply subtract 1 from the function as a whole.

• Subtracting shifts the graph downward.
• Adding shifts it upward.

Thus, the vertically shifted function becomes \(g(x) = |x+2| - 1\) which translates the graph down by 1 unit. This transformation modifies the outputs, affecting the height of the graph relative to the x-axis.