Question

Given the function \(f(x)=a b^{x-h}+k\), describe the effects of \(a, h\), and \(k\) on the graph of the function.

Short answer

In the function \(f(x) = a b^{(x-h)} + k\), the value of \(a\) controls the vertical stretch, shrink, or reflection of the graph. The value of \(h\) controls the horizontal shift of the graph, while \(k\) controls the vertical shift.

Step-by-step solution

Effects of Parameter \(a\)

The parameter \(a\) affects the vertical stretch or compression and reflections on the x-axis of the graph. If \(a > 1\), it stretches the graph vertically, if \(0 < a < 1\), it compresses the graph vertically, and if \(a\) is negative, it reflects the graph about the x-axis.

Effects of Parameter \(h\)

The parameter \(h\) affects the horizontal shift of the graph. If \(h > 0\), the graph shifts h units to the right and if \(h\) is negative (let's say -h), it shifts |h|=h units to the left.

Effects of Parameter \(k\)

The parameter \(k\) affects the vertical shifting of the graph. If \(k > 0\), the graph shifts up by k units. If \(k < 0\), the graph shits down by |k|=k units.

Key concepts

Vertical Stretch and Compression

The parameter \(a\) in a function plays a crucial role in determining how the graph is stretched or compressed vertically. A simple way to think about this is by considering how \(a\) affects the "intensity" of the graph's features.
If \(a > 1\), each point on the graph is pulled away from the x-axis, which results in a vertical stretch. This means the graph becomes taller, emphasizing height and the differences between values.
On the other hand, if \(0 < a < 1\), the graph experiences vertical compression. Points are pushed towards the x-axis, causing the graph to appear shorter and the differences between heights to be less stark.
Also, a negative \(a\) value not only stretches or compresses but also flips the graph over the x-axis. This reflection mirrors each point of the graph across the x-axis, essentially turning it upside down.

• \( a > 1 \): Vertical stretch
• \( 0 < a < 1 \): Vertical compression
• \( a < 0 \): Reflection over the x-axis, plus stretch/compression

Through these transformations, \(a\) enables you to change the graph's visual scale and orientation without shifting it around the plane.

Horizontal Shift

Horizontal shifts involve moving the entire graph left or right along the x-axis. This transformation is controlled by the parameter \(h\) in the function.
If \(h > 0\), the graph of the function moves \(h\) units to the right. Picture sliding the graph horizontally to make room for new values along the left side.
Conversely, if \(h < 0\), the graph shifts \(|h|\) units to the left. Think of this as moving the graph in the opposite direction, placing new values on the right.This shift doesn't alter the shape or size of the graph—it simply relocates it within the coordinate plane.

• \( h > 0 \): Shift right by \(h\) units
• \( h < 0 \): Shift left by \(|h|\) units

These adjustments are useful for aligning the function graph with specific data points or features of interest in a real-world scenario.

Vertical Shift

Vertical shifts adjust the graph by moving it up or down along the y-axis. This transformation is managed by the parameter \(k\).
When \(k > 0\), the entire graph lifts upwards by \(k\) units. This change increases the y-values of every point on the graph, resulting in a higher position on the plane.
If \(k < 0\), the graph descends \(|k|\) units. Each y-value decreases, causing the graph to move down.

• \( k > 0 \): Shift upward by \(k\) units
• \( k < 0 \): Shift downward by \(|k|\) units

Vertical shifts influence the baseline level of the graph, providing the flexibility to adjust where the graph resides without altering its shape or horizontal position. These transformations are significant when aligning the graph to match changes in elevation, baseline measurements, or to model phenomena with rising or falling trends.