Problem 36 - Algebra 2

Question

Without graphing either function, explain how the graph of \(y=[x+2]-3\) is related to the graph of \(y=[x+1]-1\).

Step-by-Step Solution

Answer

\(y=[x+2]-3\) is shifted 1 unit left and 2 units down, compared to \(y=[x+1]-1\).

Step 1: Identify the Basic Function

The given functions are related to the step function, often written as the greatest integer function or floor function. The basic form is \(y = [x]\), which steps at each integer value.

Step 2: Understand Transformation Rules

The function \( f(x) = [x] \) can be modified in the form \( f(x+c)-d \), where \(c\) shifts the graph horizontally and \(d\) shifts the graph vertically.

Step 3: Analyze Horizontal Shifts

For the function \( y = [x+2]-3 \), the term \( x+2 \) indicates a horizontal shift to the left by 2 units. For \( y = [x+1]-1 \), the term \( x+1 \) indicates a horizontal shift to the left by 1 unit.

Step 4: Analyze Vertical Shifts

For \( y = [x+2]-3 \), the ",-3" moves the graph down by 3 units. For \( y = [x+1]-1 \), the ",-1" moves the graph down by 1 unit.

Step 5: Determine Overall Transformation

Comparing \( y = [x+2]-3 \) and \( y = [x+1]-1 \), \( y = [x+2]-3 \) has an additional horizontal shift of 1 unit to the left and an additional vertical shift of 2 units down, compared to \( y = [x+1]-1 \).

Key Concepts

horizontal shifts

When dealing with functions like the greatest integer function, it is crucial to understand how horizontal shifts work. A horizontal shift occurs when we add or subtract a value within the argument of the function, such as inside the brackets of the floor function. For example, in the functions mentioned:

• For the function \( y = [x+2] - 3 \), the \(+2\) inside the brackets indicates a horizontal shift.
• This shift moves the entire graph to the left by 2 units because the expression \( x+2 \) implies that intake values for \( x \) must be 2 units lesser to produce the same output as \( y = [x] \).
• Similarly, the function \( y = [x+1] - 1 \) includes \(+1\) within the brackets, indicating a shift to the left by 1 unit.

Horizontal shifts are represented by the equation \( y = [x+c] \), where the value \( c \) dictates the direction and magnitude of the shift:

• \( +c \) moves the graph left.
• \( -c \) moves the graph right.

Remember, the horizontal shift does not alter the step nature of the floor function itself; it only alters the starting points of these steps.

vertical shifts

Vertical shifts are another simple transformation that can affect the graph of a function. When a constant is added or subtracted outside of the function, it results in a vertical shift. In the functions discussed:

• The term \(-3\) in \( y = [x+2] - 3 \) leads to a vertical shift downward by 3 units.
• The term \(-1\) in \( y = [x+1] - 1 \) moves the graph down by 1 unit.

Vertical shifts can generally be expressed in the form \( y = [x] - d \), where \( d \) is the vertical shift:

• \( +d \) shifts the graph up.
• \( -d \) shifts the graph down.

These adjustments result in each step of the graph changing its height, though the width of each step remains consistent. The distance between the steps themselves does not change, maintaining the staircase nature of the graph. It’s a straightforward way to move the entire graph up or down without adjusting the horizontal positioning of the steps.

floor function

The floor function, also known as the greatest integer function, is a type of step function. It is characterized by its step-like appearance on a graph. Here’s what you need to know:

• The function \( y = [x] \) outputs the greatest integer less than or equal to \( x \).
• For example, \([3.7] = 3\) because 3 is the largest integer less than 3.7.
• Similarly, \([-1.3] = -2\) as -2 is the largest integer less than -1.3.

This stepping nature is why transformations such as horizontal and vertical shifts are so interesting. They move the steps horizontally or vertically on the graph.

• These transformations don't affect the individual values within each step but simply reposition where those values occur along the axes.
• Notably, horizontal or vertical shifts don't alter the fundamental properties of the floor function—it will always "step" between integers.

Understanding these basic properties of the floor function can significantly aid in mastering more complex exercises and identifying how various transformations will visually affect the graph.