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1B. Polynomial and Rational Functions
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AP Precalculus

1.12 Transformations of functions

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Introduction

Learning objective

  • 1.12.A Construct a function that is an additive and/or multiplicative transformation of another function.

Function families

There are many types of functions, and we can group them into categories called function families. The parent function is the simplest form of the function in a particular family. That is a function where no transformations have been applied. Some of the function families are listed below:

Linear function

A function that has a constant rate of change. A linear function can be written in the form f\left(x\right) = ax + k

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  • Parent function: y=x

  • Domain: all real numbers

  • Range: all real numbers

  • Always increasing or always decreasing

  • One x-intercept

  • One y-intercept

  • Rate of change is always constant

Absolute value function

A function that contains a variable expression inside absolute value bars; a function of the form f\left(x\right) = a\left|x - h\right| + k

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  • Parent function: y=\left|x\right|

  • Domain: all real numbers

  • Has an absolute maximum or minimum

  • Has a vertical line of symmetry separating the increasing and decreasing intervals

  • 0, 1, or 2 x-intercepts

  • One y-intercept

  • Rate of change is constant; negative for half of the function and positive for the other half

Quadratic function

A polynomial function of degree 2. A quadratic function can be written in the form f\left(x\right) = a\left(x-h\right)^2+k where a \neq 0

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  • Parent function: y=x^2

  • Domain: all real numbers

  • Has an absolute maximum or minimum

  • Has a vertical line of symmetry separating the increasing and decreasing intervals

  • 0, 1, or 2 x-intercepts

  • One y-intercept

  • Rate of change is variable

Exponential function

A function that has a constant percent rate of change. An exponential function can be written in the form f\left(x\right) = ab^{\left(x-h\right)}+k where a \neq 0 and b>0

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  • Parent function: y=b^x

  • Domain: all real numbers

  • Has a horizontal asymptote

  • Always increasing or always decreasing

  • 0 or 1 x-intercepts

  • One y-intercept

  • Rate of change is a constant percent

Cubic function

A polynomial function of degree 3. A cubic function can be written in the form f\left(x\right) = a\left(x -h\right)^3+k where a \neq 0

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  • Parent function: y=x^3

  • Domain: all real numbers

  • Range: all real numbers

  • Point of inflection

  • 1, 2, or 3 x-intercepts

  • One y-intercept

  • Rate of change is variable

Square root function

A function with the variable term under a square root symbol. A square root function can be written in the form f\left(x\right) = a\sqrt{x -h}+k where a \neq 0

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  • Parent function: y=\sqrt{x}

  • Has an endpoint

  • Point of inflection

  • 0 or 1 x-intercepts

  • 0 or 1 y-intercepts

  • Rate of change is variable

  • Always increasing or decreasing
Rational function

A function that is a quotient of polynomials.

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  • Parent function: y=\dfrac{1}{x}

  • Has asymptotes

  • 0,1, or 2 x-intercepts

  • 0 or 1 y-intercepts

  • Rate of change is variable

Examples

Example 1

Determine the type of function represented by the following tables.

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f\left(x\right)-7-3151-3
Worked Solution
Create a strategy

Since the x-values in the table are increasing by 1 each time, we can determine the type of function by observing the rate of change of the outputs of the function.

Apply the idea

We can determine the rate of change by finding the differences between the outputs:\begin{aligned}-3-\left(-7\right)&=4\\1-\left(-3\right)&=4\\5-1&=4\\1-5&=-4\\-3-1&=-4 \end{aligned}

Since part of the function has a positive constant rate of change and the other part is a negative constant rate of change, these values represent an absolute value function.

Reflect and check

Plotting the points on a graph can also help us easily identify that the points represent an absolute value function.

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Worked Solution
Create a strategy

Observing the outputs of the function, we can see that they are growing at an increasing rate. \begin{aligned}12-8&=4\\18-12&=6\\27-18&=9\\40.5-27&=13.5\end{aligned}

There are only 2 types of functions we know of with a variable rate of change: a quadratic function and an exponential function. A quadratic function will have both an increasing and decreasing interval, but this table may only show an increasing interval portion of a quadratic function. So, we need to take a closer look at the rate of change and see if it fits an exponential pattern.

Apply the idea

An exponential function has a constant percent rate of change. Since the x-values increase by 1 each time, we can find this rate of change by dividing the outputs.\begin{aligned} 12\div8&=1.5\\18\div 12&=1.5\\27\div 18&=1.5\\40.5\div 27&=1.5 \end{aligned}Since the ratio of the outputs is the same, this function is exponential.

Reflect and check

Recall from Algebra 1 \text{Common ratio}=1+\text{growth rate} which gives us a decimal for the growth rate. Then, we multiply the rate by 100 to find the constant percent of growth.\begin{aligned}1.5&=1+r\\0.5&=r\\r&=50\%\end{aligned}

This means as x increases by 1, f\left(x\right) increases by 50\%.

Example 2

Determine the types of functions that are in this piecewise function.

f(x) = \begin{cases} x+4, & x \lt 0 \\ -2, & 0 \leq x \lt 4 \\ 12-x^2, & x\geq 4 \end{cases}

Worked Solution
Create a strategy

There are 3 types of functions that make up this piecewise function:

  • y=x+4

  • y=-2

  • y=12-x^2

We can use the structure of each equation to determine the function family it belongs to.

Apply the idea

The first equation is of the first degree and in the form f\left(x\right)=mx+b. In this case, m=1 and b=4. Therefore, the function that defines the interval x<0 is linear.

The second equation is in the form f\left(x\right)=c. In this case, c=-2. Therefore, the function that defines the interval 0\leq x <4 is constant.

The third equation is of the second degree and in the form f\left(x\right)=ax^2+bx+c. In this case, a=-1, b=0, and c=12. Therefore, the function that defines the interval x\geq 4 is quadratic.

Reflect and check
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If we had looked at the graph alone, we may have mistakenly classified the functions. Note that the linear function could have been part of an absolute value function, and the piece of the quadratic that is shown also appears linear.

This is why it is important to consider multiple features of the function instead of relying on a graph alone.

Idea summary

Constant, linear, and absolute value functions have a constant rate of change while quadratic and exponential functions have variable rates of change. Although the average rate of change for an exponential function varies, it grows or decays at a constant percent rate of change.

The rate of change, the structure of the equation, and the shape of the graph can help us classify the function into the correct family.

Function transformations

Exploration

Move the sliders to see how each one affects the graph. Choose different functions to compare the affects across the various graphs.

Loading interactive...
  1. How does each slider affect the graph?

A transformation of a function is a change in the position or shape of its graph. The function in any family with the simplest form is known as the parent function, and we frequently consider transformations as coming from the parent function. In the examples of transformations shown below, the parent function is shown as a dashed line.

Reflection

A transformation that produces the mirror image of a figure across a line

A reflection across the x-axis can be represented algebraically by g\left(x\right) = -f\left(x\right)A reflection across the y-axis can be represented algebraically by g\left(x\right) = f\left(-x\right)

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Reflection across the x-axis: {g\left(x\right)=-f\left(x\right)}
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Reflection across the y-axis: {g\left(x\right)=f\left(-x\right)}
Translation

A transformation in which every point in a figure is moved in the same direction and by the same distance

Translations can be categorized as horizontal (moving left or right, along the x-axis) or vertical (moving up or down, along the y-axis), or a combination of the two.

Vertical translations can be represented algebraically by g\left(x\right) = f\left(x\right) + kwhere k > 0 translates upwards and k < 0 translates downwards.

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Vertical translation of 4 units upwards: {g\left(x\right) = f\left(x\right) + 4}
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Vertical translation of 4 units downwards: {g\left(x\right)=f\left(x\right)-4}

Similarly, horizontal translations can be represented by g\left(x\right) = f\left(x - h\right) where h > 0 translates to the right and h < 0 translates to the left.

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Horizontal translation of 3 units left: {g\left(x\right) = f\left(x+3\right)}
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Horizontal translation of 3 units to the right: {g\left(x\right) = f\left(x - 3\right)}
Vertical compression

A transformation that scales all of the y-values of a function by a constant factor towards the x-axis

Vertical stretch

A transformation that scales all of the y-values of a function by a constant factor away from the x-axis

A vertical compression or stretch can be represented algebraically by g\left(x\right) = af\left(x\right)where 0 < \left|a\right| < 1 corresponds to a compression and \left|a\right| > 1 corresponds to a stretch.

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Vertical compression with scale factor of 0.5: {g\left(x\right) = 0.5f\left(x\right)}
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Vertical stretch with a scale factor of 2: {g\left(x\right) = 2f\left(x\right)}
Horizontal compression

A transformation that scales all of the x-values of a function by a constant factor toward the y-axis

Horizontal stretch

A transformation that scales all of the x-values of a function by a constant factor away from the y-axis

A horizontal compression or stretch can be represented algebraically by g\left(x\right) = f\left(bx\right)where \left|b\right| > 1 corresponds to a compression and 0 < \left|b\right| < 1 corresponds to a stretch.

For horizontal stretches and compressions, b=\dfrac{1}{\text{scale factor}}.

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Horizontal compression with a scale factor of 0.5: {g\left(x\right)=f\left(2x\right)}
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Horizontal stretch by a scale factor of 2: {g\left(x\right)=f\left(0.5x\right)}

When performing multiple transformations at once, we use the standard function notation a\cdot f\left[b\left(x-h\right)\right]+k with the correct values of a,b,h and k to apply transformations to f\left(x\right). When given a transformed function, we must convert it back to standard notation to correctly identify the transformations applied to the parent function.

Examples

Example 3

A function is shown in the graph below. Determine an equation for the function after it has been reflected across the x-axis and translated 4 units to the left.

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Worked Solution
Create a strategy

There are two main approaches we can use here:

  • Apply the transformations to the graph, then determine the equation of the final graph
  • Determine the equation of the graph shown, then apply the transformations algebraically
Apply the idea

Choosing to apply the transformations to the graph first results in the following:

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Looking at the final graph, we can see that it is a straight line which passes through the origin and has a slope of -\dfrac{1}{2}. So, an equation for this function is g\left(x\right) = -\frac{x}{2}

Reflect and check

The original function has an equation of f\left(x\right) = \dfrac{x}{2} - 2.

A reflection across the x-axis is represented by changing the signs of all the function values (i.e. multiplying throughout by -1), so this transformation results in h\left(x\right) = -\dfrac{x}{2} + 2.

A horizontal translation of 4 units to the left is represented by adding 4 directly to the x-values. So, this transformation results in

\displaystyle g\left(x\right)\displaystyle =\displaystyle -\frac{x + 4}{2} + 2
\displaystyle =\displaystyle -\frac{x}{2} - \frac{4}{2} + 2
\displaystyle =\displaystyle -\frac{x}{2} - 2 + 2
\displaystyle =\displaystyle -\frac{x}{2}

which is the same result.

Example 4

Point A\left(-3, 9\right) lies on the graph of f\left(x\right). Determine the coordinates of the corresponding point on the graph of g\left(x\right) = \dfrac{1}{3}\cdot f\left(x + 4\right).

Worked Solution
Create a strategy

We can use the given expression to determine the transformations from f\left(x\right) to g\left(x\right), then apply these transformations to the point A.

Apply the idea

In the expression \dfrac{1}{3}\cdot f\left(x + 4\right), the +4 inside of function f indicates a horizontal translation of 4 units to the left, while the \dfrac{1}{3} outside of function f indicates a vertical shrink by a factor of \dfrac{1}{3}.

Shifting the point left 4 units takes the point to \left(-7, 9\right), and shrinking it vertically by 3 results in the point \left(-7, 3\right).

Reflect and check

We can also think about these transformations algebraically.

The only point we know on the graph of f is A, which tells us that f\left(-3\right) = 7. We can rewrite this to be in the correct form for g\left(x\right) as follows:

\displaystyle f\left(-3\right)\displaystyle =\displaystyle 9Known point
\displaystyle f\left(-7 + 4\right)\displaystyle =\displaystyle 9Rewrite -3 in the form ⬚ + 4
\displaystyle \dfrac{1}{3}\cdot f\left(-7 + 4\right)\displaystyle =\displaystyle 3Multiply both sides by \dfrac{1}{3}
\displaystyle g\left(-7\right)\displaystyle =\displaystyle 3Use definition of g\left(x\right)

So, the corresponding point is \left(-7, 3\right).

Example 5

The graph of a function f\left(x\right) is shown below.

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Determine the equation after the function has been translated 6 units right and horizontally stretched by a factor of 2.

Worked Solution
Create a strategy

The graph has a v-shape, which tells us that it is an absolute value function. The parent absolute value function is f\left(x\right)=\left|x\right|. We need to give the transformed function a new name, like g\left(x\right), since we are creating a new function.

A horizontal translation and a horziontal stretch are represented by b and h in the function notation g\left(x\right)=f\left[b\left(x-h\right)\right]. Remember b=\dfrac{1}{\text{stretch factor}} for horizontal stretches and compressions.

Apply the idea

In this case, the stretch factor is 2, so b=\dfrac{1}{2}. To translate the function 6 units right, h=6. To find the equation after it has been horizontally translated and stretched, we need to find g\left(x\right)=f\left[\dfrac{1}{2}\left(x-6\right)\right].

\displaystyle g\left(x\right)\displaystyle =\displaystyle f\left[\dfrac{1}{2}\left(x-6\right)\right]
\displaystyle =\displaystyle \left|\dfrac{1}{2}\left(x-6\right)\right|
\displaystyle =\displaystyle \left|\dfrac{1}{2}x-3\right|

The equation of the transformed function is g\left(x\right)=\left|\dfrac{1}{2}x-3\right|.

Reflect and check

After we simplified the equation, the transformations that we applied to the graph are no longer easy to identify. If one is not careful, it may seem as if this function was only translated 3 units to the right. However, that is not the case as we know this function was translated 6 units to the right.

In order to identify the transformations when given an equation, we must convert it to standard form first by factoring out the coefficient of x:g\left(x\right)=a\cdot f\left[b\left(x-h\right)\right]+k

The standard form of the equation in this example is g\left(x\right)=\left|\frac{1}{2}\left(x-6\right)\right| which shows that the parent absolute value function was translated 6 units to the right and stretched horizontally by a factor of 2.

b

Graph g\left(x\right) and f\left(x\right) on the same coordinate plane.

Worked Solution
Create a strategy

A horizontal stretch will make the graph wider, and horizontal translation of 6 units to the right will move the vertex to \left(6,0\right). We will stretch the graph horizontally to determine the correct shape of the graph, then we will shift it to the right.

Apply the idea

Using a table of values for f\left(x\right) can help us stretch the function horizontally. A table of values for the parent absolute value function is:

x-4-3-2-101234
f\left(x\right)432101234

We can double the x-values and leave the y-values the same in order to stretch the graph horizontally by a factor of 2.

x-8-6-4-202468
y432101234

Using the tables to graph both functions, we can see we have stretched f\left(x\right) horizontally by 2.

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Now, we simply need to shift all the points on the stretched graph to the right 6 units.

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Reflect and check

In general, when the transformations are both horizontal transformations (or a reflection across the y-axis), apply the reflections, stretches, and compressions first. Apply the translations last. The same rule applies when the transformations are both vertical (or a reflection across the x-axis).

Example 6

Describe how g\left(x\right)=3^{x-5}-6 has been transformed from its parent function, f\left(x\right)=3^{x}.

Worked Solution
Create a strategy

When we compare f\left(x\right) and g\left(x\right), we can see that there are numbers that have been added and subtracted. Addition and subtraction represent translations.

Apply the idea

5 is subtracted from the input values and 6 is substracted from the output values. In function notation, it can be represented as:

g\left(x\right)=f\left(x-5\right)-6

The graph of g\left(x\right) is the graph of f\left(x\right) after it has been translated right 5 units and down 6 units.

Reflect and check

When we graph the functions, we can see that f\left(x\right) has been shifted right 5 units and down 6 units to obtain the graph of g\left(x\right).

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Idea summary

The reflections and translations can be summarized as follows:

A figure showing a summary of reflections and translation. The expression a f of left bracket b left parenthesis x minus h right parenthesis right bracket plus k is shown. Under the column labeled vertical: for a: when a is greater than 0, no reflection; when a is less than 0, reflection across the x-axis; when absolute value of a is greater than 1, vertical stretch; and when absolute value of a is between 0 and 1, vertical compression; for k: when k is greater than 0, vertical translation up; and when k is less than 0, vertical translation down. Under the column labeled horizontal: for b: when b is greater than 0, no reflection; when b is less than 0, reflection across the y-axis; when absolute value of b is greater than 1, horizontal compression; and when absolute value of b is between 0 and 1, horizontal stretch; for h: when h is greater than 0, horizontal translation right; and when h is less than 0, horizontal translation left.

Outcomes

1.12.A

Construct a function that is an additive and/or multiplicative transformation of another function.

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