Functions and Funky Symbols

Question 1

What is a function?

Answer 1

A function is a relationship of values which are expressed as an ordered pair in which the first number is related to exactly one other number.

Question 2

What are the four ways functions be presented within the SAT math component?

Answer 2

Functions are presented through a statement of definition, a listing of values in terms of x and f(x) in a T-graph form, a graph of values on a coordinate plane, or within a word problem.

Question 3

What steps might be necessary to solve an inequality?

Answer 3

Solve an inequality the same way you would solve an equation, i.e. simplify and isolate the variable.

*** Note: When multiplying or dividing both side of the inequality by the same negative number, it reverses the direction of the sign of the inequality.

Question 4

How does multiplying or dividing an inequality by a negative number affect the direction of the sign of the inequality?

Answer 4

Multiplying or dividing an inequality by a negative number reverses the direction of the inequality.

Example:

-2x > 10 Dividing both sides by -2

x < -5

Question 5

Find the roots.

x² - 5x + 6 = 0

Answer 5

x² - 5x + 6 = 0

The sum of the roots equals -5. The

Root 1 = -2
Root 2 = -3

Question 6

If both c and b in a quadratic equation are positive, what can you conclude about the roots of this equation?

x² + bx + c = 0

Answer 6

The roots of this equation are both negative.

Given that c is positive, the product of the roots must be positive and therefore, the roots must be the same sign.

root 1 * root 2 = c
(-root 1) * (-root 2) = c

Since the sum of the roots equals to the negative of b, you can conclude the roots of the equation are both negative.

root 1 + root 2 = -b

• Example:*
• x² + 7x* + 12 = 0

root 1 = -3 root 2 = -4

Question 7

If c is positive and b is negative in a quadratic equation, what can you conclude about the roots of this equation?

x² - bx + c = 0

Answer 7

The roots of this equation are both positive.

Given than c is positive, the product of the roots is positive. Therefore, the roots must have the same sign.

root 1 * root 2 = c
( -root 1) * ( -root 2) = c

The sum of the roots equals to the negative of b. Negative of a negative is positive. Two negative numbers cannot add up to a positive sum, therefore, the roots must be positive.

root 1 + root 2 = - (-b) = b

• Example: *
• x² - 7x* + 12 = 0

root 1 = 3 root 2 = 4

Question 8

If c is negative and b is negative in a quadratic equation, what can you conclude about the roots of this equation?

x² - bx - c = 0

Answer 8

The roots are of opposite signs and the larger root is positive.

Given than c is negative, the product of the roots is negative. Therefore, the roots must be of opposite signs.

root 1 * ( -root 2) = -c
(-root 1) * root 2 = -c

The sum of the roots equals to the negative of b. Negative of a negative is positive. The larger root of the equation must be positive.

root 1 + root 2 = - (-b) = b

• Example:*
• x² - 5x* - 6 = 0

root 1 = 6 root 2 = - 1

Question 9

If c is negative and b is positive in a quadratic equation, what can you conclude about the roots of this equation?

x² + bx - c = 0

Answer 9

The roots of this equation are of opposite signs and the larger root is negative.

Given than c is negative, the product of the roots is negative. Therefore, the roots must be of opposite signs.

root 1 * ( -root 2) = -c
(-root 1) * root 2 = -c

The sum of the roots equals to the negative of b. The larger root of the equation must be negative.

root 1 + root 2 = -b

• Example:*
• x² + 5x* - 6 = 0

root 1 = -6 root 2 = 1

Question 10

How do you use the square root method to solve a quadratic equation?

Answer 10

Isolate the squared variable on one side and have the constant on the other side. Then, take the square root of each side of the equation.

*** Remember, the square root of a number yields both a positive and a negative value.

Example:

3x² - 9 = 3 ⇒ 3x² = 12

x² = 4 ⇒ x = 2 or -2

Question 11

When should you use the square root method for solving a quadratic equation?

Answer 11

The square root method is best to use when the equation only contains x² term.

• ax^{<span>2</span>} = c*
• Example:*

(x - 4)² = 9 - take the square root of each side of the equation.

• x* - 4 = 3 ⇒ x = 7
• x* - 4 = -3 ⇒ x = 1

Question 12

What do we call a system of linear equations?

Answer 12

When you have more than one linear equation with the same set of unknowns, it’s called a system of linear equations.

The solutions must satisfy every equation in the system.

Example:

2x + 5y = 10
-5y + 3x = 15

Question 13

What methods of solving a system of linear equations do you know?

Answer 13

You can solve a system of linear equations using one of these methods:

• graphing
• substitution
• addition or subtraction

The simplest kind of a linear system involves two equations and two variables.

Question 14

How do you solve a system of linear equations with two variables using the graphing method?

Answer 14

Simply draw the graph of each equation on the same coordinate plane.

• If the lines intersect, the coordinates of the point of the intersection are the solution
• If the lines are parallel, there is no solution
• If the lines are identical (coinside), each point on the line is a solution

Question 15

How do you solve a system of two linear equations with two variables using the substitution method?

Answer 15

• Isolate the variable with the coefficient 1 in one equation
• Substitute the same variable with the resulted expression in the equation you haven’t used. Solve the equation
• Use the solution to substitute the unknown in the first equation and solve

Example:

x + 2y = 18
2x + 7y = 6

x = 18 - 2y - isolate the variable
2(18 - 2y) + 7y = 6 - substitute

x = 18 - 2y
36 - 4y + 7y = 6 ⇒ 3y = -30 ⇒ y = -10

x + 2 (-10) = 18 ⇒ x = 38 - plug in y

Question 16

When should you use the substitution method to solve a system of linear equations?

Answer 16

Use the substitution method when the coefficient of one of the variables equals 1 or -1.

Example:

x + 5y = 12
3x - 2y = 15

x = 12 - 5y
3(12 - 5y) - 2y = 15

Solve the second equation for y. Then, plug in y value into the first equation and solve for x.

Question 17

How do you solve a system of two linear equations with two variables using the addition or subtraction method?

Answer 17

• Add or subtract equations to eliminate one variable
• Solve the equation resulting from it
• Use the solution to substitute the unknown in either equation and solve

Example:

5x + 3y = 25
x - 3y = 11

Add the equations to eliminate y variable.

6x = 36 ⇒ x = 6 - plug in the x value and solve for y.

Question 18

When should you use the addition or subtraction method in a system of linear equations?

Answer 18

Use this method if the coefficients of one of the variables have the same absolute value.

*** To eliminate the variable, subtract one equation from the other if the coefficients are the same. Add the equations if the coefficients are opposite.

Question 19

Sometimes extra steps are necessary before you can use the addition or subtraction method to solve a system of linear equations.

Answer 19

Sometimes you need to rearrange the terms in the equations prior to solving them.

Example:

2(x - 3) = -6y + 5
5(y + 2) = x - 3

2x - 6 = -6y + 5
5y + 10 = x - 3

2x + 6y = 11
5y - x = -13

Now you can solve the system of equations above by using the substitution method.

Question 20

You may need to use multiplication with the addition or subtraction method to solve a system of linear equations.

Answer 20

You may need to multiply the equations to make the coefficients the same absolute value.

Example:

Evaluate the GCM of the like terms’ coefficients. It’s easier to work with 6 than with 20. Multiply the first equation by 2. Multiply the second equation by 3. Then, use the subtraction method.

3x - 5y = 19
2x - 4y = 16

6x - 10y = 38
6x - 12y = 48

2y = -10 ⇒ y = -5
2x - 4 (-5) = 16 ⇒ x = -2

Question 21

When should you multiply equations prior to using the addition or subtraction method to solve a system of linear equations?

Answer 21

Multiply if:

• The coefficient of a variable is a factor of the coefficient of the same variable in another equation
• The coefficients are relatively prime or fractional

Multiply one or both equations so that the coefficients of one of the variable will have the same absolute value.

Question 22

How many real numbers satisfy this equation?

2x² + 4 = 0

Answer 22

There are no real numbers that satisfy the equation 2x² + 4 = 0.

2x² = -4

x² = -2

No square root can be taken of a negative number within a system of real numbers.

Question 23

The squares of two integers are equal. The product of these integers could be

(a) -3,000
(b) 5,000
(c) 6,000
(d) -9,000

Answer 23

(d) -9,000

If the squares of two integers are the same, the numbers could be opposite like 30 and -30.

Question 24

The product of the roots of

(x - 15)(x + 11) = 0 is

(a) - 165
(b) 26
(c) 155
(d) 165

Answer 24

(a) -165

The roots of this equation are 15 and -11. The product must be a negative number.

Question 25

If a + b = 5 and a - b = 15, then a² - b² equals

(a) 10
(b) 20
(c) 55
(d) 75

Answer 25

(d) 75

The difference of squares formula states that

a² - b² = (a + b)(a - b) =

5 x 15 = 75

Question 26

(a + 2) divided by (4 - **a²) equals

(a) a + 2
(b) 2 - a
(c) a - 2
(d) 2a

Answer 26

(b) 2 - a

Use the difference of squares formula to factor 4 - a².

a² - b² = (a + b)(a - b)

2² - a² = (2 + a)(2 - a)

Then, reduce the resulting algebraic fraction by eliminating common factors.

Question 27

How do you evaluate an algebraic expression?

Answer 27

If you are given the values of the variables, you can evaluate an algebraic expression by plugging those values in and calculating the result.

Example:

Evaluate the algebraic expression

6 - 3x + 28 when x = -3.

Combine the like terms 6 and 28 and plug in (-3) into the expression (watch the signs).

34 - 3 (-3) = 34 + 9 = 43

Question 28

Find the sum of the two absolute values below:

IaI + I-aI = ?

(a) 2a
(b) 2 IaI
(c) IaI
(d) I-aI

Answer 28

(b) 2 IaI

The absolute value of opposite numbers is the same. Therefore, I-aI = IaI.

IaI + IaI = 2 IaI>

Question 29

Any x which satisfies below conditions also satisfies this inequality.

I2x - 5I - 7 > 0

(a) x > -6
(b) x < 7
(c) x > 6
(d) x < 1

Answer 29

(c) x > 6

Isolate the absolute value expression and solve the inequality.

I2x - 5I - 7 > 0 ⇒ I2x - 5I > 7

2x - 5 > 7 ⇒ 2x > 12 ⇒ x > 6

2x - 5 > -7 ⇒ 2x > -2 ⇒ x < -1

*** Remember, dividing by a negative number changes the direction of the inequality sign!!!

Question 30

I2x - 6I =

(a) 2x - 6
(b) 6 - 2x
(c) I6 - 2xI
(d) I2x + 6I

Answer 30

(c) I6 - 2xI

IaI = I-aI, so

I2x - 6I = I-(2x - 6)I = I-2x + 6I = I6 - 2xI

Question 31

The product of 1,562 negative numbers could equal?

(a) k
(b) -k
(c) 0
(d) -1

Answer 31

(a) k

An even number of negative factors produces a positive product. Therefore, the product of 1,562 negative numbers must be positive. Eliminate (b), (c) and (d) choices.

Question 32

If the product of 1,563 real numbers is negative, what is the least number of factors in that product that must be negative?

(a) none
(b) 1
(c) 1,561
(d) all

Answer 32

(b) 1

Only an odd number of negative factors can produce a negative product. Therefore, at least one factor must be negative.

Question 33

IIxI + yI = 6

What is the largest possible value of y?

Answer 33

The largest possible value of y is 6.

Write two equations for the absolute value.

IxI + y = 6 ⇒ IxI = 6 - y

IxI + y = -6 ⇒ IxI = -6 - y

The absolute value of x must be greater or equal to zero. In the 1st equation, y could be as large as 6. In the 2nd equation, y could be as large as -6.

Question 34

What is (are) the value(s) of x if

I2x - 5I = 11

(a) 8 or 3
(b) -3
(c) 8 or -3
(d) -8 or -3
(e) 8

Answer 34

(c) 8 or -3

By definition of the absolute value, since I2x - 5I = 11, then 2x - 5 can equal 11 or -11.

Now, solve:

2x - 5 = 11 ⇒ 2x = 16 ⇒ x = 8

2x - 5 = -11 ⇒ 2x = -6 ⇒ x = -3

Question 35

What is(are) the value(s) of 3x if

6 + I2x - 5I = 11 and x is positive?

Answer 35

x = 15

First, isolate the absolute value in the equation by subtracting 6 from each side. I2x - 5I = 5

By definition of the absolute value, since I2x - 5I = 5, then 2x - 5 can equal 5 or -5.

Now, solve:

2x - 5 = 5 ⇒ 2x = 10 ⇒ x = 5

2x - 5 = -5 ⇒ 2x = 0 ⇒ x = 0

Remember to pick only the positive value of x.

Question 36

What is one possible value of x if Ix - 5I = I-11I?

(a) -6
(b) -11
(c) -16
(d) -21

Answer 36

(a) x = -6

The absolute value of -11 is 11.

By definition of the absolute value, since Ix - 5I = 11, then x - 5 can equal 11 or -11.

Now, solve:

x - 5 = 11 ⇒ x = 16
x - 5 = -11 ⇒ x = -6

16 is not among the answer choices, so choice (a) -6 is the correct answer.

Question 37

What is the slope of a vertical line?

Answer 37

Vertical lines have no slope or an undefined slope.

Question 38

What is the slope of a horizontal line?

Answer 38

Horizontal lines have the slope of zero.

Question 39

Write:

The point-slope form of a linear equation.

Answer 39

This is the point-slope form of a linear equation:

• y - y₁* = m (x - x₁)
• m* - the slope of a line
• (x₁, y₁)* - point on the line

Question 40

How do you determine the coordinates of the point halfway between two points?

Answer 40

Use the midpoint formula:

<dl>
<dd>
<img></img>
</dd>
</dl>

where (x₁; y₁) and (x₂; y₂) are the coordinates of the end points.

***This concept will help you remember the formula. Think of it as finding the average of the x and y-coordinates of the end points.

Question 41

How do you determine the distance between two points in a coordinate graph?

Answer 41

Use the distance formula:

<dl>
<dd>
<img></img>
</dd>
</dl>

d - distance between two points with (x₁;y₁) and (x₂;y₂) coordinates.

*** This concept will help you remember the distance formula: It is a variant of the Pythagorean Theorem**. ** You can create a right triangle connecting two points to find the hypotenuse. The hypotenuse will be the distance between two points.

Question 42

What can you tell about the line described by the equation below?

x + y = -5

Answer 42

• This line crosses y-axis at a point (0, -5).
• The slope of this line is -1.
• The line has a negative slope, it falls from left to right.

Rewrite the equation in the slope-intercept form.

• y = mx + b*
• x* + y = -5 ⇒ y = - x - 5

Question 43

Find the distance between points A and B with the following coordinates?

A (-4; 3) and B (4; -3)

Answer 43

The distance between A and B equals 10.

Use the distance formula to find the answer.

(4-(-4))² + (-3-3)² = 64 + 36 = 100

Take square root of 100 and the result will be the distance between points A and B.

Question 44

What is the slope of a line parallel to the line with this equation?

3x + 6y = 18

Answer 44

The slope of the parallel line is -¹/₂.

To find out the slope of a line described by the equation, rewrite the equation in the slope-intercept form:

y = mx + b

6y = -3x + 18 ⇒ y = -¹/₂x + 3

Slopes of parallel lines are the same.

Question 45

What is the slope of a line perpendicular to the line described by the equation below?

6x + 2y = 12?

Answer 45

The slope of the perpendicular line is ¹/₃.

Put the equation into the slope-intercept form:

***y = mx + b***
2*y* = -6*x* + 12 ⇒ *y* = -3*x* + 6

The slopes of two perpendicular lines are the opposite reciprocals of one another. Since the slope of this line is -3, the slope of the perpendicular line is ¹/₃.

Question 46

a - b

Answer 46

When you add two numbers with different signs, subtract the absolute values (the smaller from the larger) and keep the sign of the number with the greater absolute value.

• Example:*
• 5 + 3 = -2

|-5| - |3| = 5 - 3 = 2

Since -5 has the greater absolute value than 3, the result is negative.

Question 47

What is scientific notation and where is it used?

Answer 47

Scientific notation is a way to express very large or very small numbers. You substitute zeros for power of 10 in exponential form.

Example:

6,000,000,000 = 6 x 10⁹ (count the number of places to the right of the first non-zero number)

0.00056 = 5.6 x 10^-4(count the number of places to the right of the decimal point, up to and including the first non-zero number)

Question 48

In order to solve a linear equation, you might need to simplify it. What operations might be necessary?

Answer 48

You may need to do all or some of these to simplify the original equation:

1. Combine similar terms within grouping symbols
2. Use distributive property
3. Remove unnecessary parentheses
4. Combine like terms

Once you’ve simplified the original equation, proceed with isolating the variable by using inverse operations.

Example:

4 (3x + 3 - 8x) - (-6x) = 26

1. 4 (3 - 5x) - (-6x) = 26
2. 12 - 20x - (-6x) = 26
3. 12 - 20x + 6x = 26
4. 12 - 14x = 26

Question 49

Define:

• Constant
• Coefficient

Answer 49

• A coefficient is a multiplicative factor of a variable
• Constant is a fixed value

Example:

5x - 2 = 8

5 is a coefficient, x is a variable, 2 and 8 are constants.