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Graphing Functions

A **transformation** is a change of position or size of a figure. When we transform a figure, we create a new figure that is related to the original. There are three different types of trasformations:

These congruent triangles are simply plotted on different parts of the coordinate plane, so only the locations of the triangles are different. To do a translation, move each point according to the given criteria.

**Example:** Given ∆ABC, translate it as follows (x + 4, y + 3)

The following are the typical types of translations that are tested on the ACT:

- To graph y = f(x) + c, shift the graph of y = f(x) upward c units.
- To graph y = f(x) - c, shift the graph of y = f(x) downward c units.

a) \(g(x)=x^{2}+3\)

b) \(h(x)=x^{2}-2\)

- To graph y = f(x - c), shift the graph of y = f(x) right c units.
- To graph y = f(x + c), shift the graph of y = f(x) left c units

**Example:** The graphs of \(f(x)=x^{2}\) is shifted to form

a) \(g(x)=(x+4)^{2}\)

b) \(h(x)=(x-2)^{2}\)

To complete a reflection, move each point according to the given criteria.

**Example:** Given ∆EFG, reflect the shape over the x-axis.

- To graph y = - f(x), reflect the graph of y = f(x) in the x-axis.
- To graph y = f(-x), reflect the graph of y = f(x) in the y-axis.

a) \(f(x)=-x^{2}\)

b) g(x) = √(-x)

A **dilation** is a transformation that enlarges or reduces a figure by a **scale factor**. The scale factor is the amount by which you stretch or shrink the original figure.

In the above figure ABC, side AB has a length of 5 units and side AC has a length of 4 units. If ABC is enlarged by a scale factor of 2 to make AB'C', side AB' has a length of 10 units and side AC' has a length of 8 units.

- When you enlarge a figure, the scale factor is greater than 1.
- When you shrink a figure, the scale factor is less than 1.

Usually, when a dilation is done on a coordinate plane, the origin (0,0) is the center of dilation. If the center of dilation is the origin, simply multiply the coordinates of the original shape by the given scale factor, k: **( x , y ) → ( xk , yk).**

Then plot your new shape.

- If c > 1,
**stretch**the graph of y = f(x) vertically by a factor of c. - If 0 < c < 1,
**shrink**the graph of y = f(x) vertically by a factor of c.

A. \(g(x)=3x^{2}\)

B. \(h(x)=x^{2}\)

- if c > 1,
**shrink**the graph of y = f(x) horizontally by a factor of 1/c. - if 0 < c < 1,
**stretch**the graph of y = f(x) horizontally by a factor of 1/c.

A **rotation** is a transformation that turns a figure around a fixed point called the **center of rotation**. The number of degrees that the figure turns is called the angle of rotation. The shape moves either **clockwise** or **counter-clockwise**.

If you measure \(\angle\)ACA', you will find that A' moved 90 ̊ in a counterclockwise direction.

So this means that ABC was rotated 90 ̊ in a counterclockwise direction to form A'B'C. Also, the two triangles are congruent - the corresponding sides are the same length and the corresponding angles are the same degrees.

**Example:** Rotate ΔABC 90 ̊ clockwise

**Example:** Rotate ΔABC 180 ̊ clockwise

- A is (3, 1), so A| is (-3, -1). Both coordinates are negative because A will be in QIII.
- B is (3, 4), so B| is (-3, -4) because B| will be in QIII.
- C is (1, 1), so C| is (-1, -1) because C| will be in QIII.

Asymptotes are quite apparent in graphs of functions. If you are only given the equation of a function, there are some techniques that can be used to identify different asymptotes.

- If there is no denominator, there are no vertical asymptotes
- If there is no denominator, set the denominator to zero
- Solve for x, that is your vertical asymptote

** Solved Example 2 **What are the vertical asymptotes of the function \(f(x)=\frac{x^{2}+1}{x-9}\)

**Solution:** Step 1: Set the denominator equal to zero.

x − 9 = 0

Step 2: Solve for x.

X = 9

This function has one vertical asymptote located at X = 9. We can see that this is the case by considering the graph of the function, shown below.

There are a few key rules to help determine if a function has a horizontal asymptote. To find the horizontal asymptotes, use the following steps:

** Step 1:** If the highest degree of power in the numerator is

** Step 2: **If the highest degree in the numerator and denominator

** Step 3:** The horizontal asymptote is at

**Example:** \(y=\frac{2x^{2}+3x-1}{4x^{2}-16}\)

Highest degree power in numerator: 2

Highest degree power in denominator: 2

**Horizontal asymptote:** \(\frac{2x^{2}}{4x^{2}}=\frac{2}{4}=\frac{1}{2}\)

This function has one horizontal asymptote at y = 1/2

**Example:** Find the horizontal and vertical asymptotes of the function \(y=\frac{x^{4}+3x}{6x^{4}-x^{2}}\)

S** tep 1: **Calculate

\(6x^{4}-x^{2}=0\)

\(x^{2}(6x^{2}-1)=0\)

\(x^{2}=0 \; or\; 6x^{2}-1=0\)

\(x=0 \; or\; 6x^{2}=1\)

\(x=\pm \frac{\sqrt{1}}{6}\)

**Vertical asymptotes**: x = 0 and \(x=\pm \frac{\sqrt{1}}{6}\)

** Step 2:** Calculate horizontal asymptotes.

Highest term in numerator: \(x^{4}\)

Highest term in denominator: \(6x^{4}\)

Since the powers on the terms are the same, the horizontal asymptote occurs at the point that we receive by \(\frac{x^{4}}{6x^{4}}=\frac{1}{6}\)****

**Horizontal asymptote: **y = 1/6

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View Resources NowA TRANSFORMATION is a change of position or size of a figure. When we transform a figure, we create a new figure that is related to the original.

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