Question

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\). \(f(x)=\left(\frac{2}{3}\right)^x ;\) reflection in the \(x\)-axis, followed by a vertical stretch by a factor of 6 and a translation 4 units left

Short answer

The rule for \(g(x)\) that represents the indicated transformations of the graph of \(f(x) = \left(\frac{2}{3}\right)^x\) is \(g(x) = -6 \left(\frac{2}{3}\right)^{(x+4)}\).

Step-by-step solution

Reflection

Reflection of the function \(f(x) = \left(\frac{2}{3}\right)^x\) in the \(x\)-axis is accomplished by multiplying the function by -1. The new function, say \(h(x)\), is: \(h(x) = - f(x) = - \left(\frac{2}{3}\right)^x\).

Vertical Stretch

A vertical stretch by a factor of 6 implies that we multiply the function \(h(x)\) by 6. Let's define the new function as \(i(x)\): \(i(x) = 6h(x) = -6 \left(\frac{2}{3}\right)^x\).

Translation

A translation of 4 units to the left is achieved by replacing \(x\) in \(i(x)\) with \(x+4\), giving us the transformed function, \(g(x)\): \(g(x) = i(x+4) = -6 \left(\frac{2}{3}\right)^{(x+4)}\).

Key concepts

Reflection in the x-axis

Reflection is a type of transformation that flips a function across a specified axis. In the context of this exercise, reflecting across the x-axis involves multiplying the entire function by -1. Essentially, this action changes the direction of each point on the graph, flipping it upside down.
This means that if a point on the graph was above the x-axis, after reflection, it will be mirrored below the x-axis.
The original function given is \(f(x) = \left(\frac{2}{3}\right)^x\). By reflecting it in the x-axis, we alter it to become \(h(x) = -\left(\frac{2}{3}\right)^x\).
It is crucial to remember that reflection across the x-axis affects the vertical position of the graph, but not its horizontal position. All positive outputs become negative, and all negatives become positive. This makes the graph look like it is flipped vertically.

Vertical Stretch by a Factor

Vertical stretching is another important transformation for functions. When a function undergoes a vertical stretch, we multiply the function by a specific factor.
This affects how narrow or wide the graph appears.
If the factor is greater than 1, the graph will stretch away from the x-axis, making it taller.
In this exercise, the function \(h(x) = -\left(\frac{2}{3}\right)^x\) is vertically stretched by a factor of 6.
This means that every point on the graph is pulled away from the x-axis by 6 times its vertical distance.
The resulting transformation is called \(i(x) = -6\left(\frac{2}{3}\right)^x\).
Keep in mind that vertical stretching affects the height of the graph, without altering the horizontal spacing between points. It's like taking a photo and pulling the top upwards.

Function Translation

Function translation involves shifting the position of the graph to a new location.
This doesn't alter the shape but merely the position of the graph along the x or y-axis.
In this case, a horizontal translation is made by moving the graph 4 units left.
To achieve this, we adjust the input \(x\) by replacing it with \(x + 4\).
This transformation takes the function \(i(x) = -6\left(\frac{2}{3}\right)^x\) and modifies it to the final function, \(g(x) = -6\left(\frac{2}{3}\right)^{x+4}\).
Horizontal translations change the x-coordinates of points on the graph, shifting everything left or right.

• Moving left means replacing \(x\) with \(x + \text{number of units}\).
• Moving right involves replacing \(x\) with \(x - \text{number of units}\).

It's important to remember that translation repositions the graph without affecting its shape.