Problem 12 - Algebra 2

Question

Graph each function as a transformation of its parent function. $$ y=-2(5)^{x+3} $$

Step-by-Step Solution

Answer

The graph starts at the point (-3, -1), decreases rapidly for positive x-values, and approaches but never reaches the x-axis as x decreases.

Step 1: Identify the values

The function can be rewritten as y = -2(5)^(x - (-3)). Thus we have A = -2, b = 5, h = -3, and k = 0. These values are derived from the formula y = A * b^(x - h) + k. The negative value of A indicates a reflection over the x-axis, b is the base of the exponent, h is the horizontal shift (shift to the left since h is negative) and as there is no added or subtracted value outside the function, k is 0 (which means there is no vertical shift)

Step 2: Plot the parent function

Start by drawing the parent function, \(y = 5^x\), without any transformations. This should be an exponential growth curve that passes through the point (0,1) and increases quickly for positive x-values.

Step 3: Apply transformations

Now apply the transformations. The reflection over the x-axis flips the graph upside down, so draw a curve that passes through the point (0,-1) and decreases quickly for positive x. The horizontal shift moves this curve three units to the left, so the point that the curve passes through is (-3,-1). The final graph should look like the parent function, y = 5^x, in terms of shape, but reflected over the x-axis and shifted three units to the left.

Key Concepts

Exponential Functions

Exponential functions are a vital mathematical concept where the variable appears as the exponent of a constant base. The general form of an exponential function is \(y = a \cdot b^x\), where \(a\) is a constant, \(b\) is the base, and \(x\) is the exponent. It describes rapid growth or decay, depending on the value of \(b\).

When the base \(b\) is greater than 1, the function represents exponential growth, like compound interest or population growth. If \(0 < b < 1\), it represents exponential decay, often seen in radioactive decay or depreciation. The exponential function is crucial because it models real-world situations where change accumulates proportionally to the current state.

The parent exponential function, specifically \(y = b^x\), is a curve that runs through the point (0,1), where \(b^0 = 1\). This point is a key characteristic as it represents the function's value when \(x\) is zero. In transformations, the behavior and location of this central point help us understand how the function shifts or flips, providing a visual point of reference.

Reflection Over the X-Axis

A reflection over the x-axis is a type of transformation that alters a graph by flipping it upside down, producing a mirror image across the horizontal axis. In mathematical terms, this transformation changes the sign of the function’s output. For a function \(y = f(x)\), its reflection would be \(y = -f(x)\).

Consider the exponential function \(y = 5^x\). When reflected over the x-axis, it becomes \(y = -5^x\). This transformation makes what was increasing now decrease, creating a downward curve that still retains the exponential nature of the original function.

This type of transformation is visually apparent in our exercise function \(y = -2(5)^{x+3}\). The negative coefficient before the exponential term (-2) flips the curve. It causes the exponential growth curve to transform into one of exponential decay. This reflection is essential in understanding negative values in the function and prepares for further transformations involving shifts.

Horizontal Shift

Horizontal shifts move the graph of a function left or right in the coordinate plane. This transformation changes only the x-values of the function, effectively translating the entire graph horizontally. For a function \(y = f(x - h)\), if \(h\) is positive, the graph shifts \(h\) units to the right. Conversely, if \(h\) is negative, the graph moves \(h\) units to the left.

In the provided exercise, the function \(y = -2(5)^{x+3}\) includes the transformation \((x + 3)\). This expression signifies a horizontal shift of three units to the left. It moves every point on the graph three places in the negative x-direction.

Understanding horizontal shifts is crucial in graph analysis. It helps us predict the new positions of points based on their initial configurations. In combination with other transformations like reflections and stretches, recognizing shifts enhances our ability to visualize and recreate complex graph transformations accurately.

Parent Function Graphing

Parent function graphing involves the initial step of plotting the simplest form of a function before any transformations are applied. This step is crucial, providing a base to understand how subsequent modifications influence the graph's behavior and position.

The parent function for our exercise example is \(y = 5^x\). This base form is an exponential growth curve, generally steep and passing through the point (0,1) due to the base raised to the power of zero. It serves as a reference point for graphing transformed functions.

Once we've plotted this curve, other transformations—such as reflections, shifts, or stretches—are applied to alter its position and shape. By starting with the parent graph, increments due to transformations become easier to comprehend.

This step ensures that each transformation's effect—whether a shift, reflection, or other—can be accurately visualized and tracked, facilitating a comprehensive understanding of graph behavior in response to mathematical changes.