Problem 11 - Precalculus: Functions and Graphs

Question

Sketch the graph of \(f\) if \(a=2\) (a) \(f(x)=a^{x}\) (b) \(f(x)=-a^{x}\) (c) \(f(x)=3 a^{x}\) (d) \(f(x)=a^{x+3}\) (e) \(f(x)=a^{x}+3\) (f) \(f(x)=a^{x-3}\) (g) \(f(x)=a^{x}-3\) (h) \(f(x)=a^{-x}\) (i) \(f(x)=\left(\frac{1}{a}\right)^{x}\) (j) \(f(x)=a^{3-x}\)

Step-by-Step Solution

Answer

Each function shifts, reflects, or stretches the base graph \(f(x) = 2^x\) in different ways. For each part, manipulate the graph of \(2^x\) accordingly.

Step 1: Understand the Base Function

The base function here is exponential: \[ f(x) = 2^x \]This is an exponential growth function where the base is 2. The graph is increasing as \(x\) increases, starting from \(y = 0.5\) when \(x = -1\), \(y = 1\) when \(x = 0\), and \(y = 2\) when \(x = 1\).

Step 2: Analyze Part (a)

For \(f(x) = 2^x\), plot the exponential function with base 2. The graph passes through points such as \((0,1)\), \((1,2)\), \((2,4)\) and so on. The curve approaches the x-axis as an asymptote to the left and rises rapidly to the right.

Step 3: Analyze Part (b)

For \(f(x) = -2^x\), flip the graph of \(f(x) = 2^x\) over the x-axis. Points on the curve include \((0,-1)\), \((1,-2)\), and \((2,-4)\).

Step 4: Analyze Part (c)

For \(f(x) = 3 \cdot 2^x\), this is a vertical stretch by a factor of 3. Multiply the y-values of every point on \(f(x) = 2^x\) by 3, resulting in points like \((0,3)\), \((1,6)\), and \((2,12)\).

Step 5: Analyze Part (d)

For \(f(x) = 2^{x+3}\), shift the graph of \(f(x) = 2^x\) horizontally 3 units to the left. A point like \((0,1)\) will shift to \((-3,1)\).

Step 6: Analyze Part (e)

For \(f(x) = 2^x + 3\), this represents a vertical shift upwards by 3 units. Shift all points on the graph of \(f(x) = 2^x\) up by 3, such as \((0,1)\) moving to \((0,4)\).

Step 7: Analyze Part (f)

For \(f(x) = 2^{x-3}\), shift the graph of \(f(x) = 2^x\) to the right by 3 units. For example, the point \((0,1)\) becomes \((3,1)\).

Step 8: Analyze Part (g)

For \(f(x) = 2^x - 3\), shift the graph of \(f(x) = 2^x\) vertically down by 3 units. This changes points like \((0,1)\) to \((0,-2)\).

Step 9: Analyze Part (h)

For \(f(x) = 2^{-x}\), reflect the graph of \(f(x) = 2^x\) over the y-axis. The curve decreases from left to right, passing through points like \((0,1)\), \((-1,2)\), and \((-2,4)\).

Step 10: Analyze Part (i)

For \(f(x) = \left(\frac{1}{2}\right)^x\), it is identical to \(f(x) = 2^{-x}\) – a decreasing exponential graph that mirrors \(f(x) = 2^x\) over the y-axis.

Step 11: Analyze Part (j)

For \(f(x) = 2^{3-x}\), rewrite as \(f(x) = 2^3 \cdot 2^{-x} = 8 \cdot 2^{-x}\). This graph is a vertical stretch of the graph in part (h) by a factor of 8, moving from points like \((0,8)\) to \((1,4)\).

Key Concepts

Graph Transformations

In mathematics, graph transformations are operations that alter the position, shape, and orientation of a graph relative to its standard or "parent" graph. When working with exponential functions, such as the base function \( f(x) = 2^x \), graph transformations allow us to adjust and explore various functional behaviors. For exponential functions, common transformations can include shifting, reflecting, and stretching or compressing the graph. Understanding these transformations involves observing how different parameters in the function equation affect the graph's appearance and position. By analyzing different transformations, students can gain a better understanding of the interplay between algebraic manipulation and graphical representation.

Vertical Shifts

Vertical shifts involve moving the graph up or down along the y-axis without altering its shape. This is achieved by adding or subtracting a constant from the function. For example, in the function \( f(x) = 2^x + 3 \), the graph of \( f(x) = 2^x \) is lifted 3 units upwards. Conversely, in \( f(x) = 2^x - 3 \), the graph shifts 3 units downwards. To visualize this transformation:

• Add a positive constant, the graph shifts up.
• Subtract a positive constant, the graph shifts down.

These vertical adjustments do not affect the x-intercepts or the horizontal asymptote of the graph, which remains at \( y = 0 \). Instead, they modify the y-values of all points on the graph accordingly.

Horizontal Shifts

Horizontal shifts move the graph to the left or right along the x-axis. The transformation is typically a result of adding or subtracting a constant within the exponent of an exponential function. For an exponential function like \( f(x) = 2^{x+3} \), the graph of \( f(x) = 2^x \) moves 3 units to the left. Meanwhile, in \( f(x) = 2^{x-3} \), the graph moves 3 units to the right. Here's how to understand the shift direction:

• Add inside the exponent, shift the graph to the left.
• Subtract inside the exponent, shift the graph to the right.

The horizontal shifts do not impact the function's growth rate or the behavior as \( x \to \infty \), but they modify where each previously plotted point appears along the x-axis.

Reflections

Reflections flip the graph over a specified axis. In the context of exponential functions, reflections occur across either the x-axis or y-axis. For instance, in \( f(x) = -2^x \), the graph of \( f(x) = 2^x \) is reflected over the x-axis, creating an upside-down mirror image. Consequently, all the y-values become their negative counterparts. Another form of reflection can be seen in functions like \( f(x) = 2^{-x} \), where the graph is reflected over the y-axis. Here, the exponential graph that originally increases becomes a decreasing one.Understanding reflections helps in observing:

• Reflection over the x-axis changes the sign of the y-values.
• Reflection over the y-axis changes the direction of growth from increasing to decreasing or vice versa.

Reflections do not change the shape of the graph, but they do modify its orientation relative to the axes.