Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x+43)\)

Step-by-Step Solution

Answer

The graph is shifted 43 units to the left.

Step 1: Understanding Function Transformations

In function transformations, changes inside the function's argument affect the x-values. Specifically, an addition of a constant to the x-variable inside a function represents a horizontal shift of the graph.

Step 2: Identifying the Transformation

The given function is \( y = f(x+43) \). The transformation involves adding 43 to the x-variable, which causes a horizontal shift.

Step 3: Determining the Direction of the Shift

Adding a positive constant \( c \) (in this case, 43) inside the function \( f(x+c) \) results in a shift to the left by \( c \) units. Therefore, the graph of \( y = f(x+43) \) is the graph of \( f(x) \) shifted 43 units to the left.

Key Concepts

Horizontal Shift

A horizontal shift refers to the movement of a graph along the x-axis, either to the left or to the right. This occurs when a constant is added or subtracted inside the function's argument. When we see a transformation in the format of \( f(x+c) \), where \( c \) is a constant, it indicates a horizontal shift.

In our original function \( y = f(x+43) \), the \(+43\) inside the function's argument is key. Although it seems like it should move right due to the positive value, it actually shifts the graph of the function \( f(x) \) to the left by 43 units. This is a crucial point to understand about horizontal shifts: seeing \( f(x+c) \) means we move the graph \( c \) units to the opposite of the sign's direction seen.

Key points about horizontal shifts include:

• Positive \( c \) inside \( f(x+c) \) shifts the graph to the left.
• Negative \( c \) inside \( f(x-c) \) shifts the graph to the right.
• The vertical position of the graph is unaffected.

Reflect on horizontal shifts as transformations that translate the graph sideways without changing its shape, reflecting its core functionality in altering the appearance of a graphed function along the x-axis.

Graph of a Function

A function graph visually represents a set of ordered pairs \((x, y)\) that satisfy the function \( y = f(x) \). This blueprint offers a wealth of information, revealing everything from continuity to intercepts.

The graph aids our understanding of transformations by illustrating how changes like shifts or stretches impact the overall picture. In particular, with horizontal shifts like \( y = f(x+43) \), the graph maintains its original shape and steepness. It simply moves sideways, allowing us to "see" the transformation as a shift in the location on the x-axis.

Observing the graph:

• Helps us identify where specific values like intercepts are now located.
• Visualizes the transformation in physical terms.
• Maintains the original shape and symmetry of the function.

By focusing on the movement, rather than the alteration of its form, graphs act as a guiding map to make sense of the mathematical changes taking place.

Function Argument

The function argument is essentially the variable component of a function—in \( y = f(x) \), it's the \( x \). This element is crucial when considering shifts like horizontal transformations since it's manipulated to create a shift in the graph's position.

When modifying a function argument, as in \( y = f(x+43) \), you're not altering the output directly but tweaking the input values the function acts upon. This change affects all x-values uniformly, guiding a coherent shift across the entire graph.

Important aspects regarding the function argument include:

• An alteration here results in direct changes to the graph's location.
• Understanding changes in arguments helps in determining how the graph moves.
• Manipulating this value allows for various types of transformations beyond just horizontal shifts.

Thus, the function argument is a pivotal part in understanding how graphical transformations influence both the function and its depiction on a coordinate plane.