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:
1. Translation
One type of transformation is a TRANSLATION, which moves all a figure's points the same distance and direction. However, the orientation and size of the figure remain the same. The newly translated figure is called the IMAGE, and the new points are written with a PRIME SYMBOL (').
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)
First, write the original coordinates. Then calculate each translated point by adding 4 units to the x-value (x + 4) and then adding 3 units to the y-value (y + 3). Lastly, plot and label the image as A'B'C'
The following are the typical types of translations that are tested on the SAT:
1. Vertical Shifts
Suppose c > 0.
- 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.
Example: The graphs of \(f(x)=x^{2}\) is shifted to form
a) \(g(x)=x^{2}+3\)
b) \(h(x)=x^{2}-2\)
2. Horizontal Shifts
Suppose c > 0.
- 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) \(h(x)=(x-2)^{2}\)
2. Reflection
A reflection is a transformation that flips a figure over a line of symmetry- if you fold the paper at the line of symmetry, the original and the image will match exactly.
To complete a reflection, move each point according to the given criteria.
Example: Given ∆EFG, reflect the shape over the x-axis.
First, count how many units each point is away from the line of symmetry (in this case, the x-axis) and draw the reflected point the same distance away on the other side. Lastly, plot and label the image as E'F'G'.
A few shortcuts regarding the reflection of graphs often covered on the SAT:
- 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.
Example: Sketch the graph of each function.
a) \(f(x)=-x^{2}\)
b) g(x) = √(-x)
Example: Given the graph of f(x) = √x, use transformations to graph f(x) = 1 - √(1+x)
3. Dilation
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.
When you change the size of a figure, there is a center of dilation, which is the fixed point in the coordinate plane around which the figure expands or contracts.
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.
AB'C', is twice as large as ABC. The two triangles are also similar because the corresponding angles are congruent and the sides are proportional.
- 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.
Example: Given the polygon EFGH, draw the dilation image with the center of dilation at the origin and a scale factor of 3.
These are the typical types of dilations that are tested on the SAT:
1. Vertical Stretching and Shrinking
To graph y = cf(x):
- 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.
Example: Sketch the graphs of \(f(x)=x^{2}\)
A. \(g(x)=3x^{2}\)
B. \(h(x)=x^{2}\)
Example: Sketch the graph of the function \(f(x)=1-2(x-3)^{2}\)
2. Vertical Stretching and Shrinking
To graph y = f(cx):
- 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.
Example: The graph of y = f(x) is shown below.
4. Rotation
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.
A rotation does not change the size or shape of the figure. This means the image after the rotation is congruent to the original figure.
If you measure \(\angle\)ACA', you will find that A' moved 90 ̊ in a counterclockwise direction.
If you measure \(\angle\)BCB', you will find that B' 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.
Rotations can also be performed on a coordinate plane. Usually, the origin (0,0) will be the center of rotation.
Example: Rotate ΔABC 90 ̊ clockwise
What happened? The x- and y-coordinates swapped places and then took the appropriate signs for quadrant I.
Example: Rotate ΔABC 180 ̊ clockwise
Rotate each point 180 ̊ or two quadrants clockwise (the coordinates swap twice).
- 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.
Example: Graph the ∆ABC 270 ̊ and its image after a rotation of 90 ̊ counterclockwise.
5. Asymptotes
Asymptotes are lines that functions approach but never reach. Not all functions have asymptotes. There are two types of asymptotes: vertical asymptotes and horizontal asymptotes.
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.
1. Vertical Asymptote
To find the vertical asymptotes, use the following steps:
- 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 \(f(x)=\frac{x^{2}+1}{x-9}\)
A. x = -9 and x = 9
B. x = −9
C. x = -3
D. 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.
2. Horizontal Asymptote
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 greater than the highest degree in the denominator, then no horizontal asymptote exists.
Step 2: If the highest degree in the numerator and denominator are equal, then a horizontal asymptote does exist.
Step 3: The horizontal asymptote is at the fraction of the terms with the highest coefficients.
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 \(y=\frac{x^{4}+3x}{6x^{4}-x^{2}}\)
Step 1: Calculate vertical asymptotes.
\(6x^{4}-x^{2}=0\)
\(x^{2}(6x^{2}-1)=0\)
\(x^{2}=0 \; or\; 6x^{2}-1=0\)
\(x=0 \; or\; 6x^{2}=1\)
\(x=\pm \frac{\sqrt{1}}{6}\)
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
