# 3.1: Transformations of f(x)

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In this section, you will practice manipulating a given graph, according to the corresponding function notation. We’ll use the function $$f$$, graphed below, for demonstration throughout this section. But any graph will do!

By the end of this section, you’ll learn to read function notation like a blueprint, reposition the graph according to the instructions it gives. You will learn how to apply these transformations.

Transformation Function Notation $$(c > 0)$$ Notes
Shift up $$c$$ units 0)\)">$$0>y = f(x) + c$$ If addition or subtraction is outside of the function, shift up or down.
Shift down $$c$$ units 0)\)">$$0>y = f(x) − c$$
Shift right $$c$$ units 0)\)">$$0>y = f(x-c)$$ If addition or subtraction is inside the function, shift left or right.
Shift left $$c$$ units 0)\)">$$0>y = f(x+c)$$
Reflection across the $$x$$-axis 0)\)">$$0>y = -f(x)$$ If the negative is outside of the function, there is an $$x$$-axis reflection.
Reflection across the $$y$$-axis 0)\)">$$0>y=f(-x)$$ If the negative is inside of the function, there is a $$y$$-axis reflection.
Vertical scale change 0)\)">$$0>y=cf(x)$$ If the multiplier, $$c$$, is outside of the function, there is a vertical scale change.
Horizontal scale change 0)\)">$$0>y=f(cx)$$ If the multiplier, $$c$$, is inside the function, there is a horizontal scale change.

The examples demonstrate the appropriate manipulation to the graph of $$f$$. The original graph $$f$$ is referred to as the parent graph. The concept can be applied to any function, but the same parent function is used in each example below.

##### Example 3.1.1

Using the graph of $$f$$, transform the graph appropriately.

1. $$y = f(x + 2)$$
2. $$y = f(x) + 2$$
3. $$y = f(x − 2)$$

Solution

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y = f(x +2) &\text{Plus \(2$$ Inside Function; Shift Left $$2$$} \end{array}\)

Step 2: Shift each point $$2$$ units left:

Step 3: Answer: $$y = f(x + 2)$$

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y = f(x) +2 &\text{Plus \(2$$ Outside Function; Shift Up $$2$$} \end{array}\)

Step 2: Shift each point $$2$$ units up:

Step 3: Answer: $$y = f(x )+ 2$$

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y = f(x -2) &\text{Minus \(2$$ Inside Function; Shift Right $$2$$} \end{array}\)

Step 2: Shift each point $$2$$ units right:

Step 3: Answer: $$y = f(x -2)$$

##### Example 3.1.2

Using the graph of $$f$$, transform the graph appropriately.

1. $$y = f(x) − 2$$
2. $$y = −f(x)$$
3. $$y = f(−x)$$

Solution

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y = f(x) -2 &\text{Minus \(2$$ Outside Function; Shift Down $$2$$} \end{array}\)

Step 2: Shift each point $$2$$ units down:

Step 3: Answer: $$y = f(x) -2$$

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y = -f(x) &\text{Negative Outside Function; \(x$$-axis Reflection} \end{array}\)

Step 2: Change each $$y$$-value to its opposite.

Step 3: Answer: $$y = −f(x)$$

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y = f(-x) &\text{Negative Inside Function; \(y$$-axis Reflection} \end{array}\)

Step 2: Change each $$x$$-value to its opposite.

Step 3: Answer: $$y = f(x) -2$$:

##### Example 3.1.3

Using the graph of $$f$$, transform the graph appropriately.

1. $$y = 2f(x)$$
2. $$y = f(2x)$$

Solution

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y = 2f(x) &\text{Times \(2$$ Outside Function; Vertical Scale Change $$2$$} \end{array}\)

Step 2: Multiply each $$y$$-value by $$2$$.

Step 3: Answer: $$y = 2f(x)$$:

Caution: Horizontal scale change is counter-intuitive! For $$y = f(cx)$$, the graph of $$f$$ is scaled horizontally by a factor of $$\dfrac{1}{c}$$. In other words, if $$c > 1$$, then the graph is compressed. If $$0 < c < 1$$, (a proper fraction) then the graph is stretched horizontally

1. Step 1: Identify the transformation on the parent graph, $$f$$.

$$\begin{array}&&y =- f(x) &\text{Minus \(2$$ Outside Function; Shift Down $$2$$} \end{array}\)

Step 2: Multiply each $$x$$-value by $$\dfrac{1}{2}$$.

Step 3: Answer: $$y = f(2x)$$:

## Combining Transformations

In many cases, graphing a function will require more than one transformation. Perform transformations in the same order as PEMDAS, the order of operations. The next example demonstrates the manipulation of a graph using multiple transformations.

##### Example 3.1.4

Using the graph of $$f$$,

Sketch the graph: $$y = −2f(x − 1) − 3$$

Solution

Since transformations are to be performed in the order of PEMDAS, each transformation is noted then ordered.

The transformations of $$4$$ points of $$f$$ are charted below. After completing all transformations, plot the transformed points stated in the final column. Connect the points to create the graph.

$$f$$'s Point Shift Right: Add one to each $$x$$-value. Vertical Stretch & Reflect: Multiply $$y$$-values by $$−2$$. Shift Down: Subtract $$3$$ from each $$y$$-value
$$(−4, 4)$$ $$(−3, 4)$$ $$(−3, −8)$$ $$(−3, −11)$$
$$(−2, −2)$$ $$(−1, −2)$$ $$(−1, 4)$$ $$(−1, 1)$$
$$(2, 1)$$ $$(3, 1)$$ $$(3, −2)$$ $$(3,-5)$$
$$(5, 0)$$ $$(6, 0)$$ $$(6, 0)$$ $$(6,-3)$$

The graph $$y = −2f(x − 1) − 3$$ is pictured below:

## Try It! (Exercises)

For exercises 1-17, use the graph of $$f$$ at right, along with the appropriate transformation(s), to sketch the transformed function. Use graph paper.

1. $$y = −f(x)$$
2. $$y = f(x) + 1$$
3. $$y = f(x + 2)$$
4. $$y = f(x − 3)$$
5. $$y = f(x) − 4$$
6. $$y = f(−x)$$
7. $$y = 3f(x)$$
8. $$y = f(2x)$$
9. $$y = \dfrac{1}{2} f(x)$$
10. $$y = f \left( \dfrac{1}{2} x \right)$$
11. $$y = −f(x)$$
12. $$y = 2f(x + 1)$$
13. $$y = 4 + f(x − 2)$$
14. $$y = 3 − f(x)$$
15. $$y = f(x − 3) − 3$$
16. $$y = −3f \left( \dfrac{1}{3} x \right)$$
17. $$y = 2f(−x) − 1$$

For #18-24, write the function that would correspond to the described transformations on $$f$$.

1. Reflect $$f$$ through the $$y$$-axis, then move the graph up $$5$$ units.
2. Stretch $$f$$ vertically by a factor of $$6$$, then shift the graph left $$8$$ units.
3. Shift the graph of $$f$$ to the right $$10$$ units, then shift the graph down $$7$$ units.
4. Reflect $$f$$ through the $$x$$-axis, move the graph $$3$$ units right, then shift the graph up $$12$$ units.
5. Reflect $$f$$ through the $$x$$-axis, stretch it vertically by a factor of $$4$$, then shift it down $$5$$ units.
6. Reflect $$f$$ through the $$y$$-axis, shift it left (9\) units, then shift it up $$8$$ units.
7. Compress $$f$$ horizontally by a factor of $$\dfrac{1}{4}$$, reflect it through the $$x$$-axis, then shift down $$1$$ unit.

This page titled 3.1: Transformations of f(x) is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.