Transformations of Functions - AP PreCalculus Study Guide

The Core Idea: Transformations of Functions

The study of function transformations provides a systematic way to understand how a family of functions is generated from a single "parent" function. By applying a specific set of parameters to a known function, $f(x)$, we can create a new function, $g(x)$, whose graph is a geometrically altered version of the original. These alterations include shifting the graph horizontally or vertically, stretching or compressing it horizontally or vertically, and reflecting it across the axes.

The general form for these transformations is encapsulated in the equation $y=af(b(x-c))+d$. Each parameter—$a$, $b$, $c$, and $d$—controls a specific type of transformation. By analyzing these parameters, we can deconstruct the relationship between the graph of the original function $y=f(x)$ and the graph of its transformation. This allows us to predict the appearance of a transformed function's graph from its equation and, conversely, to determine the equation of a function based on its graphical relationship to a parent function.

Key Rules for Transformations

The transformations of a function $f(x)$ are governed by the parameters $a,b,c,d$ in the general equation $y=af(b(x-c))+d$. These parameters can be categorized by whether they affect the function's input (horizontal effects) or output (vertical effects).

Vertical Transformations (Affecting the Output)

These transformations are controlled by the parameters $a$ and $d$, which are applied outside the function $f$.

1. Vertical Translation (Shift):

   • The parameter $d$ determines the vertical shift of the graph.

   • If $d>0$, the graph of $y=f(x)$ is translated $d$ units upward.

   • If $d<0$, the graph of $y=f(x)$ is translated $|d|$ units downward.

2. Vertical Dilation (Stretch/Compression) and Reflection:

   • The parameter $a$ determines the vertical dilation and reflection across the x-axis.

   • Dilation: The magnitude, $|a|$, controls the vertical stretching or compressing.

     • If $|a|>1$, the graph of $y=f(x)$ is stretched vertically.

     • If $0<|a|<1$, the graph of $y=f(x)$ is compressed vertically.

   • Reflection: The sign of $a$ controls reflection across the x-axis.

     • If $a<0$, the graph of $y=f(x)$ is reflected across the x-axis.

Horizontal Transformations (Affecting the Input)

These transformations are controlled by the parameters $b$ and $c$, which are applied inside the function $f$.

1. Horizontal Translation (Shift):

   • The parameter $c$ determines the horizontal shift of the graph.

   • If $c>0$, the graph of $y=f(x)$ is translated $c$ units to the right (as seen in the form $x-c$).

   • If $c<0$, the graph of $y=f(x)$ is translated $|c|$ units to the left (as seen in the form $x+|c|$).

2. Horizontal Dilation (Stretch/Compression) and Reflection:

   • The parameter $b$ determines the horizontal dilation and reflection across the y-axis.

   • Dilation: The magnitude, $|b|$, controls the horizontal stretching or compressing.

     • If $|b|>1$, the graph of $y=f(x)$ is compressed horizontally.

     • If $0<|b|<1$, the graph of $y=f(x)$ is stretched horizontally.

   • Reflection: The sign of $b$ controls reflection across the y-axis.

     • If $b<0$, the graph of $y=f(x)$ is reflected across the y-axis.

Understanding Input vs. Output Transformations

A critical nuance in understanding transformations is distinguishing between operations performed on the input ($x$) and operations performed on the output ($f(x)$). This distinction is the key to separating horizontal and vertical effects.

• Vertical Transformations ($a$ and $d$): These are output transformations. They occur after the function $f$ has been evaluated. In the equation $y=af(x)+d$, you first find the value of $f(x)$, and then you multiply by $a$ and add $d$. These operations affect the y-coordinates of points on the graph directly and intuitively. For example, adding a positive $d$ increases the y-coordinate, moving the graph up.

• Horizontal Transformations ($b$ and $c$): These are input transformations. They occur before the function $f$ is evaluated. In the equation $y=f(b(x-c))$, you first take the input x$, subtract $c$, and then multiply by $b$. This modified value is then fed into the function $f$. Because these operations alter the input required to produce a certain output, their effects on the graph can seem counter-intuitive. For instance, to get the same output as $f(x_0)$, we need the new input x$ to satisfy $b(x-c)=x_0$. Solving for $x$ shows the inverse operations are what determine the final position. This is why $x-c$ results in a shift to the right (a positive direction) and why $|b|>1$ results in a horizontal compression.

A crucial detail is the factored form $b(x-c)$. If a function is presented as $g(x)=f(bx-k)$, you must first factor out $b$ to correctly identify the horizontal shift: $g(x)=f(b(x-k/b))$. In this case, the horizontal shift is $k/b$, not $k$. The CED-specified form $y=af(b(x-c))+d$ standardizes this to avoid ambiguity.

Core Concepts & Rules

• General Form: The graph of $y=af(b(x-c))+d$ is a transformation of the parent graph $y=f(x)$.

• Vertical Shifts: The value of $d$ translates the graph vertically. A positive $d$ shifts the graph up; a negative $d$ shifts it down.

• Horizontal Shifts: The value of $c$ translates the graph horizontally. In the form $f(x-c)$, a positive $c$ shifts the graph right; a negative $c$ shifts it left.

• Vertical Dilations: The magnitude of $a$, $|a|$, dictates the vertical stretch or compression. If $|a|>1$, the graph is stretched vertically. If $0<|a|<1$, it is compressed vertically.

• Horizontal Dilations: The magnitude of $b$, $|b|$, dictates the horizontal stretch or compression. If $|b|>1$, the graph is compressed horizontally. If $0<|b|<1$, it is stretched horizontally.

• Reflection across x-axis: If $a$ is negative ($a<0$), the graph is reflected across the x-axis.

• Reflection across y-axis: If $b$ is negative ($b<0$), the graph is reflected across the y-axis.

Step-by-Step Example 1: Describing Transformations from an Equation

Problem: Let the parent function be $f(x)$. Fully describe the sequence of transformations applied to the graph of $f(x)$ to obtain the graph of $g(x)=-4f(3x-6)+5$.

Step 1: Rewrite the function in standard transformation form.

The standard form is $y=af(b(x-c))+d$. The horizontal transformations inside the function must be factored.

Original: $g(x)=-4f(3x-6)+5$

Factored: $g(x)=-4f(3(x-2))+5$

Step 2: Identify the transformation parameters $a,b,c,d$.

By comparing $g(x)=-4f(3(x-2))+5$ to the standard form, we can identify each parameter:

• $a=-4$

• $b=3$

• $c=2$

• $d=5$

Step 3: Describe the horizontal transformations based on $b$ and $c$.

• Parameter $b=3$: Since $|b|=3>1$, the graph of $f(x)$ is compressed horizontally.

• Parameter $c=2$: Since $c$ is positive, the graph is translated 2 units to the right.

Step 4: Describe the vertical transformations based on $a$ and $d$.

• Parameter $a=-4$: This parameter indicates two transformations:

  • Reflection: Since $a<0$, the graph is reflected across the x-axis.

  • Dilation: Since $|a|=4>1$, the graph is stretched vertically.

• Parameter $d=5$: Since $d$ is positive, the graph is translated 5 units upward.

Step 5: Summarize the full sequence of transformations.

The graph of $g(x)$ is obtained from the graph of $f(x)$ by a horizontal compression, a horizontal translation 2 units to the right, a reflection across the x-axis, a vertical stretch, and a vertical translation 5 units upward.

Step-by-Step Example 2: Identifying an Equation from Graphs

Problem: The graph of a function $f(x)$ passes through the point $(4,2)$. The graph of a transformed function, $g(x)$, passes through the point $(1,-7)$. Given that $g(x)=af(x-c)+d$, and the transformations include a vertical stretch, a reflection across the x-axis, a horizontal shift, and a vertical shift of 3 units down, determine the equation for $g(x)$.

Step 1: Identify known parameters from the description.

The problem states there is a vertical shift of 3 units down. This directly gives us the value of $d$.

• $d=-3$

Step 2: Use the points to determine the horizontal shift, $c$.

The x-coordinate of the point on $f(x)$ is $x_f=4$.

The x-coordinate of the corresponding point on $g(x)$ is $x_g=1$.

The horizontal shift is the difference: $x_g-x_f=1-4=-3$.

A shift of -3 means a translation 3 units to the left. In the form $f(x-c)$, a shift left by 3 units corresponds to $c=-3$.

So, the horizontal transformation is $f(x-(-3))=f(x+3)$.

Step 3: Use the points and known vertical shift to determine the vertical stretch and reflection parameter, $a$.

The transformation on the y-coordinates is given by $y_g=a\cdot y_f+d$.

We know the points $(4,2)$ and $(1,-7)$, so $y_f=2$ and $y_g=-7$. We also know $d=-3$.

Substitute these values into the equation:

$-7=a\cdot (2)+(-3)$

Step 4: Solve for $a$.

$-7=2a-3$

$-4=2a$

$a=-2$

Step 5: Verify the properties of $a$ and write the final equation.

The calculated value is $a=-2$.

• Since $a<0$, this confirms the reflection across the x-axis.

• Since $|a|=2>1$, this confirms the vertical stretch.

The value matches the problem description. Now, assemble the full equation using $a=-2$, $c=-3$, and $d=-3$.

$g(x)=-2f(x-(-3))+(-3)$

$g(x)=-2f(x+3)-3$

Using Your Calculator

A graphing calculator is an excellent tool for verifying the effects of transformations that you have determined analytically. It allows you to visualize the parent function and the transformed function on the same set of axes.

To verify the transformations from Example 1 ($g(x)=-4f(3(x-2))+5$) for a specific parent function like $f(x)=\sqrt{x}$:

1. Enter the Parent Function: In the $Y=$$ editor, enter the parent function into $Y_1$.

   • $Y_1=\sqrt{X}$

2. Enter the Transformed Function: Enter the full transformed function into $Y_2$.

   • $Y_2=-4\sqrt{(3(X-2))}+5$

3. Graph and Visually Inspect: Press GRAPH. You may need to adjust the window to see both graphs clearly. Observe the graph of $Y_2$ relative to $Y_1$.

   • Confirm that the starting point of the graph has moved from $(0,0)$ on $Y_1$ to $(2,5)$ on $Y_2$, which matches the horizontal shift right 2 and vertical shift up 5.

   • Confirm that the graph of $Y_2$ is below the horizontal line $y=5$, which is consistent with a reflection across the x-axis (relative to its new vertical position).

   • Confirm that the graph of $Y_2$ appears steeper (vertically stretched) and narrower (horizontally compressed) than $Y_1$.

4. Verify with the Table: Use the TABLE feature to check specific points.

   • A key point on $Y_1$ is $(1,1)$.

   • Apply the transformations to this point:

     • Horizontal: $x\to x/b+c\to 1/3+2\approx 2.333$

     • Vertical: $y\to ay+d\to -4(1)+5=1$

   • The new point should be $(7/3,1)$. Go to TBLSET (2nd + WINDOW) and set TblStart to $2.3333$ or scroll through the table to find $X=2.3333$. Check if the corresponding $Y_2$ value is 1. This provides numerical confirmation of your analytical work.

AP Exam Quick Hit

Common Question Types

• Describing Transformations from an Equation: Given an equation like $g(x)=\frac{1}{2}f(-(x+1))-4$, you will be asked to list the specific transformations that map the graph of $f(x)$ to the graph of $g(x)$.

  • Example: "Describe the sequence of transformations that maps the graph of $f(x)=|x|$ to the graph of $g(x)=-|2x-8|+3$." (This requires factoring first).

• Finding the Equation from a Description or Graph: You will be shown the graph of a parent function $f(x)$ and a transformed function $g(x)$ and asked to write the equation for $g(x)$ in terms of $f(x)$. Alternatively, you might be given a description in words.

  • Example: "The graph of $f(x)$ is reflected across the y-axis, then compressed horizontally, then shifted 4 units up to produce the graph of $g(x)$. Write an equation for $g(x)$ in terms of $f(x)$."

• Tracking a Point: Given the coordinates of a point on the graph of $f(x)$, you will be asked to find the coordinates of the corresponding point on the graph of a transformed function $g(x)=af(b(x-c))+d$.

  • Example: "If the point $(6,-2)$ lies on the graph of $y=f(x)$, what are the coordinates of the corresponding point on the graph of $y=3f(2(x-1))+5$?"

Common Mistakes

• Incorrect Horizontal Shift: For a function like $g(x)=f(2x+6)$, stating the horizontal shift is 6 units to the left. The correct approach is to factor first: $g(x)=f(2(x+3))$, revealing the shift is 3 units to the left.

• Confusing Horizontal Dilation Effects: Mistaking $|b|>1$ as a horizontal stretch and $0<|b|<1$ as a horizontal compression. The effect is the inverse of the vertical case.

• Mixing up Reflections: Incorrectly identifying $-f(x)$ as a reflection across the y-axis or $f(-x)$ as a reflection across the x-axis. Remember: operations outside the function are vertical (x-axis reflection), and operations inside are horizontal (y-axis reflection).

• Incorrect Order of Operations: When tracking a point $(x,y)$, applying translations before dilations. For the transformation $y=af(x)+d$, the correct order for the y-coordinate is to multiply by $a$ first, then add $d$.

• Sign Errors with Translations: Confusing the signs for horizontal shifts. $f(x-3)$ is a shift to the right, while $f(x+3)$ is a shift to the left.