Problem 59 - Algebra 2

Question

Describe how the graph of each function relates to the graph of \(y=4(0.8)^{x}\) $$ y=4(0.8)^{x+1}-5 $$

Step-by-Step Solution

Answer

The graph of the function \(y=4(0.8)^{x+1}-5\) is the graph of the base function \(y=4(0.8)^x\), moved left by one unit and down by five units

Step 1: Identify and Analyze the Base Function

The base function mentioned in the question is \(y=4(0.8)^x\). If it's plotted on a graph, it would reveal a decreasing exponential function that intercepts the y-axis at 4, since 4 is the y-intercept.

Step 2: Analyzing the Transformed Function

The transformed function is \(y=4(0.8)^{x+1}-5\). The exponent is written as \(x+1\) and the function is subtracted by 5. The \(x+1\) inside the exponent means there is a horizontal shift 1 unit to the left, since the transformation \(x+1\) or \(x+k\) results in the graph of the function being shifted k units to the left. The subtraction by 5 at the end of the function means there's a downward shift of the graph by 5 units, as any addition or subtraction outside of the function translates the graph vertically. Specifically, \(y-k\) shifts the graph k units down.

Step 3: Describing the Relationship

In conclusion the graph of \(y=4(0.8)^{x+1}-5\) is the graph of \(y=4(0.8)^x\), but shifted one unit to the left and five units downwards. As such, the function will still have the same declining behaviour as the base function, but its y-intercept will be at -1 (originally at 4, now moved down by 5 units) and the shift left means the general function is getting to its declining state faster.

Key Concepts

Graph Transformations

Graph transformations allow us to manipulate a graph's structure in various ways. These transformations are crucial in understanding how different components of a function affect its graphical representation. In the case of exponential functions like the one in the exercise, transformations include operations like shifts and stretches.

Transformations can be categorized mainly into:

• Horizontal Shifts: These involve moving a graph left or right.
• Vertical Shifts: These involve moving a graph up or down.
• Reflections: Flipping a graph across an axis.
• Stretches/Compressions: Altering the graph's steepness or wideness.

For the given exercise, the transformations involved are horizontal and vertical shifts. Understanding these shifts helps in predicting the function's behavior without necessarily plotting the graph.

Horizontal Shifts

Horizontal shifts impact the x-values of a function and result in the graph moving left or right on the x-axis. This shift is determined by changes inside the function’s exponent. In the function \[y=4(0.8)^{x+1}-5\]the "+1" in the exponent means the graph is shifted horizontally. This happens because when we add a positive number to x, such as "x+1", it results in a shift to the left.

Here's a straightforward way to visualize and remember horizontal shifts:

• If you see "+k" in the exponent, the graph shifts "k" units to the left.
• If you see "-k", the graph shifts "k" units to the right.

The transformation \[x+1\]not only shifts the graph but also changes when the exponential function decrease starts. The shift to the left indicates the function's behavior in its downward trend happens earlier than it would in the untransformed original.

Vertical Shifts

Vertical shifts occur when the entire graph of a function moves up or down along the y-axis. This is typically caused by additions or subtractions outside the main part of the function. In our example,\[y=4(0.8)^{x+1}-5\]we see a "-5" at the end, which implies the graph is vertically shifted downward by 5 units.

Vertical shifting is intuitive:

• A negative constant at the end of the function moves the graph downward.
• A positive constant moves the graph upward.

Thus, subtracting 5 from the original function effectively lowers all points on the graph by 5 units. As a result, the y-intercept, initially at 4 in the base function, is relocated to -1 in the transformed graph. Besides changing the y-intercept, vertical shifts don't affect the function's rate of growth or decline; they simply reposition the graph vertically along the y-axis.