## SAT SUBJECT TEST MATH LEVEL 2

## PART 2

## REVIEW OF MAJOR TOPICS

## CHAPTER 2

Geometry and Measurement

• Coordinate Geometry

• Three-Dimensional Geometry

2.1 Coordinate Geometry

### TRANSFORMATIONS AND SYMMETRY

**T**he three types of transformations in the coordinate plane are translations, stretches/shrinks, and reflections. Translation preserves the shape of a graph, only moving it around the coordinate plane. Translation is accomplished by addition. Changing the scale (stretching and shrinking) can change the shape of a graph. This is accomplished by multiplication. Finally, reflection preserves the shape and size of a graph but changes its orientation. Reflection is accomplished by negation.

Suppose *y* = *f(x*) defines any function.

• *y* = *f(x*) + *k* translates *y* = *f(x*) *k* units vertically (up if *k* > 0; down if *k* < 0)

• *y* = *f(x* – *h*) translates *y* = *f(x*) *h* units horizontally (right if *h* > 0; left if *h* < 0)

**EXAMPLES**

**1. Suppose y = f(x) = e^{x} . Describe the graph of y = e^{x} + 3.**

In this example, *k* = 3, so *e ^{x}* + 3

*= f(x*) + 3. As shown in the figures below, each point on the graph of

*y*=

*e*+ 3 is 3 units above the corresponding point on the graph of

^{x}*y*=

*e*.

^{x}**2. Suppose y = x^{2}. Describe the graph of y = (x + 2)^{2}.**

In this example *h* = –2, so (*x* + 2)^{2 }= *f(x* + 2). As shown in the figures below, each point on the graph of *y* = (*x* + 2)^{2 }is 2 units to the left of the corresponding point on the graph of *y* = *x*^{2}.

• *y* = *af(x*) stretches (shrinks) *f(x*) vertically by a factor of |*a*| if |*a*| > 1(|*a*| < 1).

• *y* = *f(ax*) shrinks (stretches) *f(x*) horizontally by a factor of if |*a*| > 1(|*a*| < 1).

**3. Suppose y = x – 1. Describe the graph of y = 3(x – 1).**

In this example |*a*| = 3, so 3(*x* + 1) = 3*f(x*). As shown in the graphs below, for *x* = 1.5 the *y*-coordinate of each point on the graph of *y* = 3(*x* – 1) is 3 times the *y*-coordinate of the corresponding point on the graph of *y* = *x* – 1.

**4. Suppose y = x^{3}. Describe the graph of** .

In this example, . As shown in the graphs below, for *y* = 0.729 the *x*-coordinate of each point on the graph of is 2 times the *x-* coordinate of the corresponding point on the graph of *y* = *x*^{3}.

• *y* = –*f(x*) reflects *y* = *f(x*) about the *x* -axis. (The reflection is vertical.)

• *y* = *f* (–*x*) reflects *y* = *f(x*) about the *y*-axis. (The reflection is horizontal.)

**5. Suppose y = ln x . Describe the graphs of y = –ln x and y = ln(–x).**

As shown in the graphs below, the graph of *y* = –ln *x* is the reflection of the graph of *y* = ln *x* about the *x* -axis, and the graph of *y* = ln (–*x*) is the reflection about the *y*-axis.

Translations, stretching/shrinking, and reflections can be combined to produce functions. Vertical transformations occur when adding, multiplying, or negating takes place after the function is applied (i.e., to *y*). The order in which multiple vertical transformations are executed does not matter. Horizontal transformations occur when adding, multiplying, or negating takes place before the function is applied (i.e., to *x*). These transformations must be taken in the following order: reflect; change the scale; then translate. Moreover, the scale factor |*a*| must be factored out of a translation.

**6. Suppose y = f(x). Use words to describe the transformation y = f (–ax + b).**

Observe that all of these transformations are horizontal. First we have to write –*ax* + *b* as . The *x*-coordinate of a point (*x*,*y*) on the graph of *y* = *f(x*) goes through the following sequence of transformations: reflected about the *y*-axis; horizontally shrunk by a factor of or stretched by a factor of |*a*|; and translated to the right.

**7. Suppose y = sin x . Describe the sequence of transformations to get the graph of y = sin(–2x + 1).**

Observe that all transformations are horizontal. Write the function as . Consider the point (4,–0.7568 . . .) on the graph of *y* = sin *x* . First reflect this point about the *y*-axis to (–4,–0.7568...). Then shrink by a factor of to (–2,–0.7568...). Then translate units right, to (–1.5,–0.7568...). The screens below show *Y*_{1} = sin *x* and *Y*_{2} = sin(–2*x* + 1) and a table showing the points (4,–0.7568...) for *Y*_{1} and (–1.5,–0.7568...) for *Y*_{2}.

Reflections about the *x* - and *y*-axes represent two types of symmetry in a graph. Symmetry through the origin is a third type of graphical symmetry. A graph defined by an equation in *x* and *y* is symmetrical with respect to the

• *x* -axis if replacing *x* by –*x* preserves the equation;

• *y*-axis if replacing *y* by –*y* preserves the equation; and

• origin if replacing *x* and *y* by –*x* and –*y*, respectively, preserves the equation.

These symmetries are defined for functions as follows:

• Symmetry about the *y*-axis: *f(x*) = *f* (–*x*) for all *x* .

• Symmetry about the *x* -axis: *f(x*) = –*f(x*) for all *x* .

• Symmetry about the origin: *f(x*) = –*f* (–*x*) for all *x* .

As defined previously, functions that are symmetric about the *y*-axis are even functions, and those that are symmetric about the origin are odd functions.

**8. Discuss the symmetry of f(x) = cos x .**

Since cos *x* = cos (–*x*), cosine is symmetric about the *y*-axis (an even function).

However, since cos *x* –cos *x* and cos (–*x*) –cos *x*, the cosine is not symmetric with respect to the *x* -axis or origin.

**9. Discuss the symmetry of x^{2} + xy + y^{2} = 0.**

If you substitute –*x* for *x* or –*y* for *y*, but not both, the equation becomes *x*^{2} – *xy* + *y*^{2} = 0, which does not preserve the equation. Therefore, the graph is not symmetrical with respect to either axis. However, if you substitute both –*x* for *x* and *–y* for *y*, the equation is preserved, so the equation is symmetric about the origin.

**EXERCISES**

1. Which of the following functions transforms *y* = *f(x*) by moving it 5 units to the right?

(A) *y* = *f(x* + 5)

(B) *y* = *f(x* – 5)

(C) *y* = *f(x*) + 5

(D) *y* = *f(x*) – 5

(E) *y* = 5*f(x*)

2. Which of the following functions stretches *y* = cos (*x*) vertically by a factor of 3?

(A) *y* = cos(*x* + 3)

(B) *y* = cos(3*x*)

(C)

(D) *y* = 3 cos *x*

(E)

3. The graph of *y* = *f(x*) is shown.

Which of the following is the graph of *y* = *f* (–*x*) – 2?

(A)

(B)

(C)

(D)

(E)

4. Which of the following is a transformation of *y* = *f(x*) that translates this function down 3, shrinks it horizontally by a factor of 2, and reflects it about the *x* -axis.

(A) *y* = –2*f(x* – 3)

(B) *y* = *f* (–2*x)* – 3

(C)

(D) *y* = –*f* (2*x*) – 3

(E)