5.05 Transformations of functions

Complete the table of values.

To fill out the first column for we need to substitute the x-values into f\left(x-1\right). After simplifying the expression in parentheses, we can substitute the value into f\left(x \right) and look for patterns.

Substituting x=-3 into f\left(x-1\right), we get \begin{aligned}f\left(-3-1\right)&=f\left(-4\right)\\&=3^{-4}\\&=\frac{1}{3^4}\\&=\frac{1}{81}\end{aligned}

Substituting x=-2 into f\left(x-1\right), we get f\left(-2-1\right)=f\left(-3\right). From the table that was given, we know f\left(-3\right)=\frac{1}{27}. All we need to do is move the element in the previous column and previous row into the next cell.

If we keep substituting values of x into f\left(x-1\right), we'd see the rest of the values can be found in the previous row and column of the original table.\begin{aligned}x&=-1, f\left(-1-1\right)=f\left(-2\right)\\x&=0,f\left(0-1\right)=f\left(-1\right)\\x&=1,f\left(1-1\right)=f\left(0\right)\\x&=2,f\left(2-1\right)=f\left(1\right)\\x&=3,f\left(3-1\right)=f\left(2\right)\end{aligned}

Let's repeat our process to see if any patterns continue. Substituting x=-3 into f\left(x-2\right), we get \begin{aligned}f\left(-3-2\right)&=f\left(-5\right)\\&=3^{-5}\\&=\frac{1}{3^5}\\&=\frac{1}{243}\end{aligned}

Substituting x=-2 into f\left(x-2\right), we get f\left(-2-2\right)=f\left(-4\right). From our work above, we know f\left(-4\right)=\frac{1}{81}. Again, we can move the element in the previous column and previous row into the next cell.

Again, the pattern we found before continues.\begin{aligned}x&=-1, f\left(-1-2\right)=f\left(-3\right)\\x&=0,f\left(0-2\right)=f\left(-2\right)\\x&=1,f\left(1-2\right)=f\left(-1\right)\\x&=2,f\left(2-2\right)=f\left(0\right)\\x&=3,f\left(3-2\right)=f\left(1\right)\end{aligned}

To verify the pattern we saw when completing the first two columns, we will repeat the process one more time. Substituting x=-3 into f\left(x-3\right), we get \begin{aligned}f\left(-3-3\right)&=f\left(-6\right)\\&=3^{-6}\\&=\frac{1}{3^6}\\&=\frac{1}{729}\end{aligned}Substituting x=-2 into f\left(x-3\right), we get f\left(-2-3\right)=f\left(-5\right). From our work above, we know f\left(-5\right)=\frac{1}{243}. Again, we can move the element in the previous column and previous row into the next cell. Since the pattern repeats, we will use it to fill in the rest of the column.

Describe how the original function is transformed when values are subtracted from x.

To determine what happened when we subtracted values from x while building the table, we can focus on what happened to individual points. An easy point to focus on is the y-intercept of the original function, \left(0,1\right).

After subtracting 1 from x, the point \left(0,1\right) moved to \left(1,1\right). This shows that each value moved to the right 1 unit.

When we subtracted 2 from x, the point \left(0,1\right) moved to \left(2,1\right). This shows that each value moved to the right 2 units. The patten continues, so the y-intercept of the original function moved to the right 3 units when we subtracted 3 from x.

Analyzing what has happened to the rest of the points, we can see that each of the y-values have been shifted to subsequent x-values, translating each point to the right. So for f\left(x-c\right), f\left(x\right) is translated c units to the right.

We can see this transformation clearly if we draw the graphs of the functions:

Reflect f\left(x\right) across the x-axis.

All the points on the parent graph are above the x-axis. To refect the function across the x-axis, we need to move each point below the x-axis. Each point should be the same distance below the axis as they were above the axis.

The point \left(-4,4\right) is 4 units above the x-axis. After reflecting the point, it should be 4 units below the x-axis.

Similarly, the point \left(-2,2\right) is 2 units above the x-axis. After reflecting the point, it should be 2 units below the x-axis.

Doing the same for each point on the graph, we get the blue graph shown below.

Create a table of values for f\left(x\right) and its reflection, g\left(x\right).

Graphing both functions on the same coordinate plane can help us observe the points on the graphs easier.

Use the table to create an equation for g\left(x\right).

To create an equation for g\left(x\right), we can use f\left(x\right) as a starting point. We will compare the outputs of both functions to determine what has changed, then apply the change to the equation of f\left(x\right).

When comparing the outputs of both functions, we can see that the y-values of f\left(x\right) are all positive while the y-values of g\left(x\right) are all negative. In other words, the outputs of g\left(x\right) can be obtained by multiplying the outputs of f\left(x\right) by -1. In function notation, this can be represented by g\left(x\right)=-1\cdot f\left(x\right).

Therefore, the equation for g\left(x\right) is \begin{aligned}g\left(x\right)&=-1\cdot f\left(x\right)\\&=-1\cdot\left|x\right|\\&=-\left|x\right|\end{aligned}

The club members spent \$45 on face-painting supplies. Write the function P\left(x\right) that represents their profit.

Profit is calculated by subtracting cost from the revenue.

P\left(x\right)=8.5x-45

Graph R\left(x\right) and P\left(x\right) on the same coordinate plane.

R\left(x\right) has a y-intercept at \left(0,0\right), and P\left(x\right) has a y-intercept at \left(0,-45\right). They both have the same slope of m=\dfrac{8.5}{1}. We can graph both functions by plotting the y-intercepts and using the slope to find other points for each line.

To graph the functions using the slope, we can rewrite the slope as m=\dfrac{8.5}{1}=\dfrac{17}{2}. This allows us to work with whole number values when counting the rise and the run.

Describe the transformation applied to R\left(x\right) to get P\left(x\right).

We have subtracted 45 from the revenue function which can be represented by P\left(x\right)=R\left(x\right)-45. Also, the graphs have the same slope which makes them parallel. The only difference between them are their y-intercepts, meaning a translation has occurred.

R\left(x\right) has been translated down 45 units to get P\left(x\right).

In context, this means that the revenue is decreased by the costs of the supplies, and that is how we get the profit function.

Describe how f\left(x\right) has been transformed to get g\left(x\right).

The outputs of f\left(x\right) and g\left(x\right) are not the same. If we can find a relationship between the outputs, then we will know that a vertical stretch or compression was applied.

The outputs of g\left(x\right) are \frac{1}{3} of the outputs of f\left(x\right). This means a vertical compression by a factor of \frac{1}{3} has been applied to f\left(x\right) to get g\left(x\right).

Create an equation for g\left(x\right).

In part (a), we found that the outputs of f\left(x\right) were multiplied by \frac{1}{3} to get the outputs of g\left(x\right). In function notation, this can be written as g\left(x\right)=\frac{1}{3}f\left(x\right) Now, we need to analyze the table from part (a) to determine the equation of f\left(x\right).

Looking at the outputs of f\left(x\right), we can see that they have an average rate of change of -1 or 1. This means it is an absolute value function. The vertex is at the origin, so f\left(x\right)=\left|x\right|.

Therefore, the equation for g\left(x\right) is

Graphing both functions on the same coordinate plane will help us clearly see the transformation and check our work.

Describe the type of transformation that transforms f\left(x\right) to g\left(x\right).

Both functions have the same y-intercept which tells us a translation has not taken place. We can look at the x- and y-values of other key points and build a table of values to help describe which transformation has taken place.

First, we can look at the change in y-values by identifying points on the graph that have the same x-values. If there has been a vertical stretch or compression, the y-values would have been multiplied by the same number.

We now know that a horizontal transformation was applied, but we still need to analyze the x-values of the functions to determine how they changed.

This shows that a horizontal compression by a factor of \frac{1}{2} was applied to f\left(x\right) to get g\left(x\right).

Write an equation for g\left(x\right) in terms of f\left(x\right).

In part (a), we found that the x-values of g\left(x\right) were \frac{1}{2} the x-values of f\left(x\right). However, the equation will be written in terms of the outputs, so we need to determine the relationship between the inputs and outputs of both functions.

Building a table of values can help us understand what is happening to the outputs.

Looking at the table, we see that g\left(1\right)=4. To get the equivalent output from f\left(x\right), x would need to be 2. In other words, the input of g\left(x\right) must be multiplied by 2 to get the equivalent output from f\left(x\right). g\left(1\right)=f\left(2\cdot 1\right) We will look at the next output of g\left(x\right) to determine if this relationship continues. The next point is g\left(2\right)=16. To get the equivalent output from f\left(x\right), x would need to be 4. In other words, the input of g\left(x\right) must be multiplied by 2 to get the equivalent output from f\left(x\right). g\left(2\right)=f\left(2\cdot 2\right)

Because the pattern continues, this relationship can be generalized as g\left(x\right)= f\left(2\cdot x\right).

By analyzing the outputs on the graph of f\left(x\right), we can determine the equation is f\left(x\right)=2^x. This means the equation for g\left(x\right) is

This equation can also be written as g\left(x\right)=4^x.