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 SAT:
a) \(g(x)=x^{2}+3\)
b) \(h(x)=x^{2}-2\)
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.
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.
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.
A. \(g(x)=3x^{2}\)
B. \(h(x)=x^{2}\)
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
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.
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.
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
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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.
The term quadratic comes from the Latin word quadratus meaning square because the variable gets squared (e.g., \(x^{2}\) ). It is also called an equation of degree 2 because of the 2 superscripts on the x.
Students struggle with function-based questions on the SAT. This chapter will give you all the information and solved examples you need to get the function questions correct
The Absolute Value of a number is its distance from zero (on the number line). Thus, an absolute value is always positive. We indicate absolute value by putting two bars around the number.
There are a few additional algebra topics often tested on the SAT. Expect each of the following to show up at least once on each SAT Math Test.
A polynomial is an algebraic expression comprised of more than two terms, usually of like bases and different powers. Polynomials frequently include integer terms as well.
Linear equations are equations in which all variables have an exponent of 1 and whose graph is a line.
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