## Ex 1: use the vertical line test to determine if a graph

Many factors influence how a river flows, including the size of the river, the amount of water it holds, the types of objects floating in the river, whether or not it is raining, and so on. These other variables must be considered if you want to accurately explain its flow. This is where a system of linear equations can improve.
A system of linear equations is made up of two or more linear equations made up of two or more variables that are all considered at the same time. Any application of mathematics contains systems of equations. They’re useful for figuring out and explaining how behaviors and processes are related. It’s uncommon to see a traffic flow pattern that is solely influenced by the weather. Accidents, the time of day, and major sporting events are just a few of the other factors that influence traffic flow in an area. We’ll look at some fundamental concepts for graphing and defining the intersection of two lines that make up a system of equations in this section.

## Consistent independent, dependent and inconsistent

Where does the graph of y = (x + a)(x + b) intersect the x-axis in the xy-plane?

### How to graph y = 3x + 2

(1) a + b equals -1 (2) At the point where the graph intersects the y-axis (0, -6) The value(s) of (x) for (y=0) are the x-intercepts of the function (f(x)), or in our case the function (graph) (y=(x+a)(x+b)). The question essentially asks you to find the roots of the quadratic equation ((x+a)(x+b)=0). ((x+a)(x+b)=0) –> (x2+bx+ax+ab=0) –> (x2+(a+b)x+ab=0) –> (x2+(a+b)x+ab=0). We know the value of (a+b) from Statement (1), but we don’t know the value of (ab) to solve the equation. When (x=0), statement (2) informs us the y-intercept point, or the value of (y) –> (y=(x+a)(x+b)=(0+a)(0+b)=ab=-6). We know the value of (ab), but not the value of (a+b), so we can’t solve the equation. We can solve the quadratic equation, which will be the x-intercepts of the given graph, since we know the values of both (a+b) and (ab). C is the correct answer. I hope that was obvious.
Are you a newcomer to the GMAT CLUB Forum? Posting Guidelines: VERBAL | QUANTITATIVE Resources and Guides: Everything You Need for Quant Questions’ Banks and Collection: QUANTITATIVE | VERBAL | Ultimate GMAT Quantitative Megathread | All You Need for Quantitative Questions’ PS: Standard deviation | Tough Problem Solving Questions with Solutions | Probability and Combinations Questions with Solutions | Tough and Tricky Exponents and Roots Questions | Bakers’ Dozen | Algebra collection | Mixed Questions | Fresh Meat Tough and tricky exponents and roots queries | The Discreet Charm of the DS | Devil’s Dozen!!! | Number Properties package | New DS set GMAT Club’s Complete Questions’ Bank is MIXED. | GMAT Probability Questions | SEVEN SAMURAI OF 2012 | Tricky questions from previous years. | Special Questions’ Directory | Best Of The Best Of 2017 | The Best Of Quant of 2016 | 20 toughest and 20 best questions of 2015 | 15 best topics of 2015 | SEVEN SAMURAI OF 2012 What are GMAT Club Tests, and what do they entail? Extra-difficult quantitative tests for cutting-edge analytics

### Graph y = 2x + 3

When interest is compounded, the amount of interest paid to principal is greater than the investment account’s annual percentage rate. As a result, the annual percentage rate, which is the very definition of nominal, does not always equate to the actual interest received.
The answers will differ. Example of a response: For a number of years, forest A’s population would rise faster than forest B’s, but since forest B’s population increases at a faster pace, it will gradually surpass forest A’s and remain so as long as the population growth models hold. Drought, an epidemic that wipes out the population, and other environmental and biological factors are some of the factors that could affect the exponential growth model’s long-term validity.
Without bounds, either increases or decreases. The horizontal asymptote of an exponential function indicates the function’s value limit as the independent variable becomes incredibly large or small.

### The graph of y=ax^2+bx+c

Starting with functions of two independent variables, we’ll illustrate what a function of more than one variable is. Identifying the domain and range of such functions, as well as learning how to graph them, are all part of this process. We also look at how to link the graphs of three-dimensional functions to the graphs of more common planar functions.
The definition of a two-variable function is very similar to the definition of a one-variable function. The key difference is that we map ordered pairs of variables to another variable rather than mapping values of one variable to values of another variable.
Each ordered pair ((x,y)) in a subset (D) of the real plane (R2) is mapped to a unique real number z by a function of two variables (z=(x,y)). The set (D) is referred to as the function’s domain. As shown in Figure (PageIndex1), the range of (f) is the set of all real numbers z that have at least one ordered pair ((x,y)D) such that (f(x,y)=z).
a. This is an example of a two-variable linear function. Since there are no values or combinations of (x) and (y) that cause (f(x,y)) to be undefined, (f(x,y)) has the domain (R2). To figure out the range, start by deciding on a z value. We must solve the equation (f(x,y)=z,) or (3x5y+2=z.) Setting (y=0) yields the equation (3x+2=z), which is one such solution. For any value of z, the solution to this equation is (x=dfracz23), which gives the ordered pair (left(dfracz23,0right)) as a solution to the equation (f(x,y)=z) (z\). As a result, the function’s range is all real numbers, or (R\).