Quant 2

Question 1

When the base of an exponential expression is a positive proper fraction (in other words, a fraction between 0 and 1), an interesting thing occurs: as the exponent increases, the value of the expression decreases!

Answer 1

(3/4) > (9/16) > (27/64)

Question 2

Given that z2 – 10z + 25 = 9, what is z?

Answer 2

{2, 8}: Since you recognize that the left-hand side of the equation is a perfect square quadratic, you will factor the left side of the equation first, instead of trying to set everything equal to 0.

Question 3

Quadratic Formula: For any quadratic equation of the form ax2 + bx + c = 0, where a, b, and c are constants, the solutions for x are given by:

Answer 3

x = - b -+ √b2 -4ac / 2a

***b2 -4ac > 0, then the square root operation yields a positive number. The quadratic formula produces TWO ROOTS of the quadratic equation. This means that the parabola crosses the x-axis twice and has two-intercepts.

***b2-4ac = 0, then the square root operation yields 0. The quadratic formula only produces ONE ROOT of the quadratic equation. This means that the parabola touches the x-axis only once and has just one-intercept.

***b2-4ac < 0, then the square root operation cannot be performed. This means that the quadratic formula produces NO ROOTS of the quadratic equation and the parabola never touches the x-axis (it had no x-intercepts).

Question 4

DIRECT PROPORTIONALITY means that the two quantities always change by the same factor and in the same direction. For instance, tripling the input will cause the output to triple as well. Cutting the input in half will also cut the output in half.

y = kx

Answer 4

x is the input value
y is the output value
k is the proportionality constant

y÷x = k which means that the ratio of the output and input values is always constant.

Question 5

INVERSE PROPORTIONALITY means that the two quantities change by reciprocal factors. Cutting the input in half will actually double the output. Tripling the input will cut the output to one-third of its original value.

y = k÷x

Answer 5

Inverse proportionality relationships are of the form

y = k÷x

where x is the input value, y is the output value, and k is the proportionality constant. This equation can also be written as yx = k, which means that the product of the output and input values is always constant.

Question 6

Linear Growth

y = mx + b

linear growth (or decay), that is, they grow (or decline) at a constant rate. Such quantities are determined by the linear function

Answer 6

In this equation, the slope m is the constant rate at which the quantity grows.

The y-intercept b is the value of the quantity at time zero, and the variable (in this case, x) stands for time. You can also use t to represent time.

For instance, if a baby weighs 9 pounds at birth and gains 1.2 pounds (the constant rate) per month, then the baby’s weight can be written as:

W = 1.2t + 9, where t is the baby’s age in months. Note that t = 0 represents the birth of the baby.

Question 7

Graph nonlinear functions of X

Answer 7

The following statements lie at the heart of all problems involving graphs of other nonlinear functions, as well as lines and parabolas.

1. If a point lies on the graph, then you can plug its coordinates into the equation y = f(x). Conversely, if a value of x and a value of y satisfy the equation y = f(x), then the point (x, y) lies on the graph of f (x).
2. To find x-intercepts, find the values of x for which y = f(x) = 0
3. To find y-intercepts, set x = 0 and find y = f(0)