Problem 10 - 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) \(y=f(x-4)+\frac{3}{4}\) (b) \(y=f(x+4)-\frac{3}{4}\)

Step-by-Step Solution

Answer

(a) Shift right 4, up \(\frac{3}{4}\); (b) Shift left 4, down \(\frac{3}{4}\).

Step 1: Understanding Subtraction Inside the Function

The expression inside the function, \(f(x-4)\), indicates a horizontal shift. Specifically, subtracting 4 implies that the graph of \(f\) shifts 4 units to the right.

Step 2: Understanding Addition Outside the Function

The expression outside the function, \(+\frac{3}{4}\), indicates a vertical shift. Adding \(\frac{3}{4}\) means that the graph moves \(\frac{3}{4}\) units upward. Thus, the graph of \(y=f(x-4)+\frac{3}{4}\) is shifted 4 units right and \(\frac{3}{4}\) units up from the graph of \(f\).

Step 3: Understanding Addition Inside the Function

In the function expression \(f(x+4)\), adding 4 inside the brackets corresponds to a horizontal shift to the left. Specifically, \(f(x+4)\) shifts the graph 4 units to the left.

Step 4: Understanding Subtraction Outside the Function

The term \(-\frac{3}{4}\) outside the function results in a vertical shift downward. Subtracting \(\frac{3}{4}\) lowers the graph by \(\frac{3}{4}\) units. Thus, the graph of \(y=f(x+4)-\frac{3}{4}\) is shifted 4 units left and \(\frac{3}{4}\) units down from the graph of \(f\).

Key Concepts

Horizontal Shift

A horizontal shift in a function graph's position occurs when adjustments are made inside the function's argument. For example, in the expressions like \(f(x-4)\) or \(f(x+4)\), the values subtracted or added to \(x\) indicate a shift along the x-axis.

• Rightward Shift: If you see \(f(x-4)\), the negative sign with the number suggests that the graph of \(f(x)\) is moved 4 units to the right. It’s a bit counterintuitive since subtracting seems like it should decrease, but here it means shift to the right, away from zero.
• Leftward Shift: On the flip side, \(f(x+4)\) means adding a number results in the graph shifting 4 units to the left. This might be odd, but adding pushes the graph left, towards smaller x-values.

Understanding these shifts can help you position the graph accurately based on changes in the function's input.

Vertical Shift

A vertical shift affects the graph's position along the y-axis. This shift results from numbers added to or subtracted from the entire function, rather than its input. Consider terms like \(+\frac{3}{4}\) or \(-\frac{3}{4}\) outside the function.

• Upward Shift: In a function like \(f(x) + \frac{3}{4}\), the addition means every point of the graph moves \(\frac{3}{4}\) units up. This essentially increases each y-coordinate by this amount, raising the entire graph.
• Downward Shift: Conversely, if you subtract like \(f(x) - \frac{3}{4}\), then the graph moves \(\frac{3}{4}\) units down. Each y-value decreases by \(\frac{3}{4}\), lowering the graph.

Vertical shifts modify the function’s output directly, changing how it stretches above or below the x-axis.

Function Transformations

Function transformations involve changes made to the graph that affect its shape, position, or orientation. The two main transformations discussed here, horizontal and vertical shifts, showcase simple yet powerful ways to manipulate a graph:

• Horizontal Shifts: By changing the input, like \(x - 4\) or \(x + 4\), the entire graph slides left or right.
• Vertical Shifts: Altering the output with terms like \(+\frac{3}{4}\) or \(-\frac{3}{4}\) shifts the graph upward or downward.

These transformations help in repositioning the graph while maintaining the original function's overall shape. By strategically leveraging these shifts, one can solve various algebraic and graphical problems, aligning the function to meet specific criteria or match certain models.