Problem 53 - College Algebra with Corequisite Support

Question

Explore and discuss the graphs of \(f(x)=4^{x},\) \(g(x)=4^{x-2},\) and \(\quad h(x)=\left(\frac{1}{16}\right) 4^{x} .\) Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left(\frac{1}{b^{n}}\right) b^{x} \quad\) for any real number \(n\) and real number \(\quad b>0\).

Step-by-Step Solution

Answer

The graph of \(b^x\) is shifted vertically by \(n\) units in \(\left(\frac{1}{b^n}\right) b^x\).

Step 1: Understanding the Function

The function \(f(x) = 4^x\) is an exponential function with a base greater than 1, so it increases as \(x\) increases. The graph passes through the point (0,1) because any number raised to the power of 0 is 1.

Step 2: Shifting the Graph Horizontally

For \(g(x) = 4^{x-2}\), the graph of \(f(x)\) is shifted two units to the right. This happens because the \(-2\) in the exponent subtracts from \(x\), causing the graph to start increasing from a later point on the \(x\)-axis.

Step 3: Scaling the Graph Vertically

The function \(h(x) = \left(\frac{1}{16}\right) 4^x\) scales the graph of \(f(x)\) vertically by a factor of \(\frac{1}{16}\). This does not affect the base of the exponent, so the shape of the graph is similar to that of \(f(x)\) but lower because of the multiplication by \(\frac{1}{16}\).

Step 4: Conjecture Formation

To form a conjecture about \(b^x\) and \(\left(\frac{1}{b^n}\right) b^x\), observe that \(\left(\frac{1}{b^n}\right) b^x = b^{x-n}\), which results in shifting the graph of \(b^x\) down by \(n\) units. The graphs are identical in shape but displaced vertically by affecting the starting position for \(y\).

Key Concepts

Graph Shifts

When dealing with exponential functions, graph shifts are a common transformation. In the exercise, comparing the graph of
\(g(x) = 4^{x-2}\) with \(f(x) = 4^x\) shows us how horizontal shifts occur. A graph shift happens when you modify the function's argument, in this case, the exponent.

- For \(g(x) = 4^{x-2}\), the \'-2\' in the exponent shifts the graph 2 units to the right. This is because every \(x\) value of \(f(x) = 4^x\) now affects the graph 2 steps earlier, essentially translating
the entire graph horizontally.

This is a key principle of transformations for exponential functions:

• \(b^{x-c}\) translates the graph \(c\) units to the right.
• \(b^{x+c}\) shifts it \(c\) units to the left.

These shifts do not change the overall shape of the graph, just the starting point in the horizontal direction.

Vertical Scaling

Another crucial transformation in exponential functions is vertical scaling. This change can be observed in the function
\(h(x) = \left(\frac{1}{16}\right) 4^x\). The term \(\frac{1}{16}\) scales the graph of \(f(x) = 4^x\) vertically.

Vertical scaling involves multiplying the entire function by a constant factor. This constant affects the graph by changing the height:

• If the factor is greater than 1, the graph stretches upwards.
• When the factor is between 0 and 1, like \(\frac{1}{16}\), it compresses the graph downwards.

The shape of the curve remains the same, retaining the characteristic of exponential growth or decay dictated by the base, but potentially appearing flatter or steeper depending
on the scaling factor. Thus, the key takeaway is that vertical factors modify the amplitude of the graph without altering its fundamental directional trend.

Exponent Laws

Exponent laws form the backbone of manipulating exponential functions. In the given exercise, these laws help us make a conjecture
about expressions like \(\left(\frac{1}{b^n}\right) b^x\). The simplification to \(b^{x-n}\) follows these core principles:

• Multiplying terms with the same base adds their exponents: \(b^{m} \times b^{n} = b^{m+n}\).
• Dividing terms with common bases subtracts the exponents: \(\frac{b^m}{b^n} = b^{m-n}\).

In the exercise, by recognizing \(\left(\frac{1}{b^n}\right) b^x = b^{x-n}\), you see the shift
effect from exponent subtraction. The result is a graph identical in shape to \(b^x\) but shifted down vertically
on the y-axis by \(n\) units. This understanding of exponent laws allows for predictions on how different algebraic manipulations
of the exponents will impact the graphs of exponential functions.