Problem 11 - College Algebra

Question

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 of \(y = f(x + 43)\) is shifted 43 units to the left of \(f(x)\).

Step 1: Identify the Transformation

The expression given is \(y = f(x + 43)\). Here, \(f(x + 43)\) is a modified version of the original function \(f(x)\). The added \(+43\) within the function argument indicates that this is a horizontal transformation.

Step 2: Determine the Type of Transformation

In the expression \(f(x + 43)\), the \(+43\) inside the parentheses affects the \(x\)-coordinate. A positive number inside the function argument indicates a shift to the left along the \(x\)-axis.

Step 3: Determine the Direction and Magnitude of Shift

Adding \(+43\) to the \(x\) inside \(f(x)\) means that every point on the graph of \(f(x)\) will move 43 units to the left. This is because \(x + 43\) implies compensating for x's increase by moving left.

Step 4: Conclusion about the Transformation

Thus, the graph of the function \(y = f(x + 43)\) is a horizontal shift of the graph of \(f(x)\), and it moves 43 units to the left.

Key Concepts

Horizontal Shift

In function transformations, a horizontal shift happens when we add or subtract a constant inside the function's argument, affecting the graph's position along the x-axis. Consider the function given as \(y = f(x + 43)\). Here, "+43" indicates how the graph of the function will shift horizontally compared to the original function \(f(x)\).

• When you see a value being added inside the function, such as \(f(x + 43)\), it represents a shift to the left.
• This might seem counterintuitive, but think of it like needing to reach the value 0 sooner; thus, we shift every input by moving left.
• The opposite happens when you subtract a value, for example, in \(f(x - 43)\), resulting in a shift to the right.

Each point of the graph moves as per this transformation. When you add 43 inside the function, every point on \(f(x)\) moves 43 units left. Grasping this concept of horizontal shifts helps in visualizing how the function changes.

Graph of a Function

The graph of a function is a visual representation of how a function behaves. It shows all possible pairs \(x, f(x)\). Understanding how various transformations affect the graph can help you predict changes without plotting every point.

• A horizontal shift moves the entire function graph horizontally.
• The graph does not change its shape, but only its position.
• For instance, in \(y = f(x + 43)\), every point moves left by 43 units.

This movement maintains the graph's integrity, ensuring that its appearance and the vertical distances between points stay unaffected. Observing how the graph shifts can give deeper insights into a function's behavior and characteristics.

Algebraic Expressions

Algebraic expressions are like the rules or formulas that define a mathematical expression. For functions, the way we manipulate these expressions dictates how transformations occur. In \(f(x + 43)\), the expression inside the parentheses causes a change in how we interpret x values.

• Think of algebraic expressions as instructions on moving and reshaping graphs.
• In transformations, operations inside the expression guide shifts.
• Every operation within an expression adjusts how x inputs are processed.

By learning how to decipher algebraic expressions, you can uncover the hidden instructions on shifting function graphs. This understanding is crucial in both problem-solving and graphing functions accurately.