Problem 29 - Algebra 2

Question

Assume that \(a\) is positive and \(b \geq 1\) Describe the effects of \(c<0, c=0\) and \(c>0\) on the graph of the function \(y=a b^{c x} .\)

Step-by-Step Solution

Answer

For \(c = 0\), the graph transforms into a horizontal line at \(y = a\). For \(c > 0\), the ascending rate of the graph increases, making it steeper. For \(c < 0\), the graph is reflected over the x-axis and descends from left to right.

Step 1: Understand the basic graph

The primary function is in the form of \(y = a \cdot b^{cx}\), which is the form of an exponential function. For \(a > 0\) and \(b \geq 1\), the basic graph of an exponential function \(y = b^x\) rises from left to right.

Step 2: Effect of \(c = 0\)

If \(c = 0\), then the function becomes \(y = a \cdot b^0\). Since any number (except 0) to the power of 0 is 1, the function turns into \(y = a\). This results in a horizontal straight line passing through \(y = a\).

Step 3: Effect of \(c > 0\)

For \(c > 0\), the function takes the form \(y = a \cdot b^{cx}\), which is an exponential function with rate \(c\). As \(c\) increases, the growth of the exponential function accelerates, making the ascent from the x-axis steeper.

Step 4: Effect of \(c < 0\)

In the case where \(c < 0\), the function transforms into \(y = a \cdot b^{-cx}\). This inverts the graph, forming a reflection on the x-axis. The graph still passes through (0, a), but falls from left to right, instead of rising.

Key Concepts

Graph Transformations

Graph transformations involve changing the position, shape, or size of a graph.
In exponential functions such as \(y = ab^{cx}\), transformations can affect how the graph behaves as values of \(x\) change.

• Vertical Shifts: If you add or subtract a constant, it moves the graph up or down.
• Horizontal Shifts: By adding or subtracting a value within the exponent, the curve shifts left or right.
• Stretching or Compressing: Multiplying \(a\) or \(c\) stretches or compresses the graph vertically or horizontally.

Understanding these transformations helps in predicting the graph's behavior with different parameter values.

Effect of Parameters on Graphs

Parameters in functions determine the specific characteristics of its graph. In \(y = ab^{cx}\), each part controls different features of the graph:

• \(a\) influences the vertical stretch and starting point of the graph. It determines the y-intercept, specifically as \( (0, a) \).
• \(b\) , the base of the exponent, sets how fast the graph rises or falls. If \(b > 1\), the graph rises as \(x\) increases; if \(0 < b < 1\), it falls.
• \(c\) affects the rate of increase or decrease. Larger \(c\) amplifies the effect of \(b\), impacting how quickly the graph ascends or descends.

These parameters control the exponential curve's steepness, direction, and starting point.

Power of Zero

The power of zero is a special case in exponential functions. When exponent \(c=0\), the function becomes very straightforward:

• Any non-zero number raised to the power of zero equals 1. Therefore, \(b^0 = 1\).

Substituting into the function \(y = a \cdot b^0\) simplifies it to \(y = a \cdot 1 = a\).
This makes the graph a horizontal line at \(y = a\), signifying no growth or decay regardless of changes in \(x\). It's a stable and constant value linked to the parameter \(a\).

Reflection of Graphs

Reflection occurs when the sign of the exponent's coefficient \(c\) changes. When \(c < 0\), it flips the graph over the x-axis:

• The function becomes \(y = ab^{-cx}\). This reflects the usual upward curve to a downward curve.
• Graphically, the curve starts from the same point, \((0, a)\), but instead of rising, it declines from left to right.

This reflection changes the graph's direction, illustrating how negative exponent coefficients impact geometric transformations.