Problem 6 - Precalculus Mathematics for Calculus

Question

Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\). (a) \(y=f(x+7)\) (b) \(y=f(x)+7\)

Step-by-Step Solution

Answer

(a) Shift the graph 7 units left; (b) Shift the graph 7 units up.

Step 1: Understanding Horizontal Shifts

The function given is \(y = f(x+7)\). This represents a horizontal shift. A horizontal shift occurs when a constant is added/subtracted to the input variable \(x\). In this case, \(x\) is replaced by \(x+7\), meaning the graph of \(f\) is shifted 7 units to the left. The \(+7\) inside the function causes the graph to move in the opposite direction of the sign.

Step 2: Understanding Vertical Shifts

The function given is \(y = f(x) + 7\). This represents a vertical shift. A vertical shift occurs when a constant is added/subtracted to the entire function value \(f(x)\). In this case, adding 7 to \(f(x)\) results in every point of the graph moving 7 units upward. The graph maintains its shape but is adjusted vertically.

Key Concepts

Horizontal Shifts

Horizontal shifts involve moving a function graph left or right along the x-axis. Imagine you have the graph of a function, and you pick it up and slide it sideways. When you see a function like \(y = f(x + 7)\), the "+7" is telling you to move the graph 7 units left.

Why left, even though it's "+7"? That's because the transformation \(x + 7\) affects the inputs of the function by shifting them oppositely. If it was \(y = f(x - 7)\), it would shift 7 units to the right.

A brief checklist for horizontal shifts:

• "\(x + a\)" means shift left by "a" units.
• "\(x - a\)" means shift right by "a" units.

It's all about how the input of the function is modified.

Vertical Shifts

Vertical shifts are all about moving the graph of a function up or down along the y-axis. If you have a function like \(y = f(x) + 7\), the "+7" indicates that the entire graph moves up 7 units. Imagine you're lifting the whole graph straight up without changing its shape.

Vertical shifts keep the function's horizontal alignment the same, so you're only adjusting the height.

Here's what to remember for vertical shifts:

• Adding a constant (e.g., "+b") moves the graph up "b" units.
• Subtracting a constant (e.g., "-b") moves the graph down "b" units.

These shifts are straightforward since they affect the output of the function.

Function Graph

The graph of a function is a visual representation of all its possible output values for every input value. It's crucial in precalculus because it helps understand the behavior of functions without crunching numbers for each point.

Transformations like shifts are modifications that give insight into the relationship between input and output. Depending on the transformation, the graph reveals different properties or behaviors of a function.

• Horizontal transformations adjust how the input is processed.
• Vertical transformations modify the result of the function.

Understanding these shifts leads to a deeper comprehension of how functions operate visually.

Precalculus Mathematics

Precalculus mathematics serves as the bridge between algebra and calculus. It involves an understanding of how functions behave under various transformations, including shifts and stretches.

These transformations help explore the world of functions more robustly, giving students tools to predict changes and understand results without performing complex calculations repeatedly.

• Horizontal and vertical shifts provide insights into translating function graphs.
• Learning these concepts is fundamental for tackling more complex calculus problems later on.

Precalculus paves the way for understanding not just mathematical relationships, but also how real-world situations can be modeled mathematically.