GRE Math: Coordinate Planes

What is the equation of the graph of a circle?

(x - a)^2 + (y - b)^2 = r^2 The center of the circle is at the point (a,b) and r > 0 Ex. an equation of a circle is: (x - 6)^2 + (y +5)^2 = 9 The center of the circle would be at (6, -5) and the radius would be 3 (because the square root of 9 is 3).

How do you find the points at which two graphs of functions intersect (points at which the two functions equal one another)?

-Set the equations of the functions equal to one another, and then isolate the variable x on one side. -Eliminate any fractions. -Set the new equation equal to 0 and solve the equation for the variable x. -The value(s) of x is/are the inputs and also the x-coordinate(s) for the point(s) of intersection. -Plug the x value(s) into one of the original functions to get the output(s) which is/are the y-coordinates for the points of intersection.

How would you solve for the exact y-intercept of a line on a coordinate plane when given two points on the line?

1. First, calculate slope using the two points given (y2 - y1 / x2 - x1). 2. Next, substitute the x and y values of any point on the line (use one of the points given) into the equation of the line (y = mx + b). 3. Solve for b, which is the y-intercept.

If two points on a plane share the same coordinates except for the signs, what can be said about their geometric relation to one another? Ex. a point J has the coordinates (4,2). Points M (4, -2), K (-4,2) and L (-4, -2) also lay on the plane with J. What is each point's relation to J?

1. M is the reflection of J about the x-axis and M and J are symmetric about the x-axis. 2. K is a reflection of J about the y-axis and J and K are symmetric about the y-axis. 3. L is the reflection of J about the origin, and J and L are symmetric about the origin.

What are the 3 possible relationships between two linear equations representing lines on a coordinate plane?

1. The equations represent two lines that intersect at some point. 2. The equations represent two lines that are parallel and do not intersect, which means that there is NO pair of numbers that satisfies both equations at the same time. 3. The equations actually represent the same line, meaning that infinitely many points along the line satisfy the two equations, one of which must be the other equation in disguise.

What is a piece-wise defined function?

A function written using two or more expressions. An absolute value function is a piece-wise defined function because when a variable is expressed as an absolute value, the value represented by the variable could be either negative or positive. Ex. When f(x) = IxI, if x is a positive number x => 0. If x is a negative number, x<0. The graph of this function would be v-shaped and consists of two linear pieces, y = x and y = -x. These pieces are joined at the origin.

What is the equation of every horizontal line on a coordinate plane?

A horizontal line has a slope of 0 since the rise is 0 for any two points on the line. Therefore, the equation for every horizontal line is y = b where b is the y-intercept.

What relationship of the form y= mx + b represent?

A line. m= slope, or rise/run. b= y-intercept.

When given the equation of a line and asked which line from the answer choices of equations of lines would intersect with the given line, what indicators do you look for (slope and y-intercept)?

Any line with a slope greater than the given line and a y-intercept less than than the given line will will intersect the given line. Look for the answer choice with the equation that reflects this situation. Disregard any answer choices that give the equation of a line with the same slope as the given line. This would indicate that the lines are parallel.

What is the easiest way to begin graphing a line on a coordinate plane?

Begin by plotting the y-intercept (b).

How do you calculate the slope of a line?

Calculate the change in Y over the change in X: (y2 - y1 / x2 - x1) Ex. If you are given the points (0,6) and (-4,0), you would calculate (6-0)/(0+4)= 6/4= 3/2. m= 3/2

When considering quadrants on a coordinate plane, in which direction are the quadrants numbered I, II, III, IV?

Counterclockwise! Top right quadrant= I Top left quadrant= II Bottom left quadrant= III Bottom right quadrant= IV

What is the equation of every vertical line on a coordinate plane?

Every vertical line has a slope that is UNDEFINED because the run would be 0 for any two points on the line, and slope (rise/run) cannot be defined when run=0 (cannot divide by 0). Therefore, the equation for every vertical line would be x = a where a is the x-intercept.

True or false: the steeper the line on a coordinate plane, the smaller the slope.

False. The steeper the line, the larger the slope because slope= rise/run. For example, a line with a slope of 2 (or 2/1) has a larger rise than run than a line with a slope of 1/2, so it will be steeper than the line with the smaller slope.

In what form should you think of the slope of a line?

Fractional form (rise/run), even if the slope is an integer (while number). Rise= change in y and run= change in x. A slope that is an integer should be thought of as that integer over 1 (ex. m=5 means m=5/1).

What is finding the point of intersection of two lines on a coordinate plane equivalent to?

It is equivalent to solving a system of two linear equations. You can find the intersection by using algebra more easily than by graphing the two lines.

How is the y-intercept of a line expressed?

It is expressed using the ordered pair (0, y). This means that the y-intercept is the point on the line at which x=0. NOT y=0!

How is the x-intercept of a line expressed?

It is expressed using the ordered pair (x,0). This means that the x-intercept is the point on the line at which y=0. NOT x=0!

What does it mean when two lines intersect on a coordinate plane?

It means that at the point of intersection (x,y), BOTH equations representing the lines are true. The ordered pair that represents the point of intersection solves BOTH equations.

What can you determine about a line with the equation y = x?

It passes through the origin, has a slope of 1 and makes a 45-degree angle with each axis. For any point with coordinates (a,b), the point with interchanged coordinates (b,a) is the reflection of (a,b) about the line y = x (points are symmetric about the line y = x). Interchanging x and y in the equation of any graph yields another graph that is the reflection of the original graph about the line x = y. Y = x becomes a line of symmetry for the original graph and its reflection.

What does slope tell you about a line on a plane?

It tells you how steep the line is AND whether the line is rising or falling.

What does the y-intercept (b) tell you about a line on a plane?

It tells you where the line crosses the y-axis.

When are two lines on a coordinate plane parallel, and when are they perpendicular?

Lines are parallel is they have the SAME SLOPE and perpendicular when their SLOPES ARE NEG. RECIPROCALS OF EACH OTHER. *Look at slope in the lines' equations and ignore y-intercept.

How do you find the y-intercept or the x-intercept when given the equation of a line?

Plug in 0 for x in the equation to find the y-intercept or plug in 0 for y in the equation to find the x-intercept. NOTE: You do not have to rearrange the equation of the line to y=mx+b form before plugging in 0 and solving for x or y.

How do you solve for the y-intercept of a line when you know the equation of the line?

Substitute 0 for x in the equation and solve for y.

How can you find the exact x-intercept of a line on a coordinate plane?

Substitute 0 for y in the equation of the line and solve for x. *If you are not given the complete equation of the line, calculate slope and find the y-intercept before substituting for 0 for y.

What would the graph of f(x) = IxI + 2 look like?

The graph of this function would be the graph of f(x) = IxI shifted up 2 units on the y-axis. It would be a v-shaped graph with two pieces joined at 2 on the y-axis or (0,2).

What would the graph of f(x) = (x + 1)^2 look like?

The graph of this function would be the graph of y = x^2 shifted 1 unit to the left on the x-axis. *Note that 1 is positive in the expression (x + 1) but would be negative as a solution, so the graph would be shifted to the left on the x-axis. If the expression was (x - 1), the graph would be shifted to the right 1 unit.

What shape is the graph of the function f(x) = the square root of x?

The graph of this function would be the upper half of a parabola on its side. It is half of the parabola y = x^2.

How will the graph of a function be affected if the function is being multiplied by a negative number?

The graph will be reflected (flipped) on the x-axis.

How will the graph of a function being multiplied by a positive number c ne affected if 0< c < 1 (in others words, c is a fraction), as in the function f(x) = x^2/4?

The graph will be shrunk vertically by a factor of c.

How will the graph of a function being multiplied by a positive number c be affected if c > 0, as in the function f(x) = 2 Ix - 1I?

The graph will be stretched vertically by a factor of c.

If the equation of a line is quadratic, what does that tell you about the line?

The line is a curve. X or Y has two values that make the equation true, meaning that there will be two X or Y intercepts, and therefore the line will result in a curve.

If a line has a negative slope, what can you determine about the direction of the line?

The line will be going DOWN and to the RIGHT/up and to the left. The run will always be plotted by moving the point to the right. a negative slope does not mean you should plot points down and the left, which might be your first instinct. A negative slope means the rise is negative, not the run.

If a line has a positive slope, can you determine about the direction of the line?

The line will be going UP and to the RIGHT/down and to the left.

If a line has a slope less than 1 but greater than 0 (0<m<1), what can you determine about the line?

The line will have a gentle slope (closer to horizontal). The rise is less than the run.

If the slope of a line is greater than 1 (m>1), what can you determine about the line?

The line will have a steep slope (closer to vertical). The rise is greater than the run.

How do you find the vertex of a parabola in the form y = ax^2 + bx + c?

The vertex (h,k) is found by computing h = -b/2a and k= (4ac - b^2)/4a

What is the relationship between two perpendicular lines on a coordinate plane?

Their slopes are negative inverses one another.

What is true about the slope of parallel lines?

They have the same slope.

What shape is the graph of the function f(x) = the negative square root of x?

This graph is the lower half of a parabola on its side. It is half of the parabola y = x^2.

When solving for the values of x that a graph of a quadratic equation of a line would touch the x-axis, how will your answer differ from the way you would answer a question about the values of x that make a quadratic equation true (factoring)?

When answering a question that involves solving a quadratic equation through factoring, you would give the values of x that are the opposite of the values you factored (ex. 0=(x+2)(x-3), the values of x would be -2 and 3), but when you give the values of x for which a graph would touch the x-axis, you would give the values as they are in the factored equation (ex. (x+2)(x-3), the answers would be 2 and -3).

How can you calculate the distance between two points on a coordinate plane?

You can use the pythagorean theorem: 1. Draw a right triangle connecting the two points. 2. Find the length of the two legs of the triangle by calculating the rise and run. 3. Use the pythagorean theorem to calculate the length of the diagonal, which is the distance between the points. NOTE: When determining how many points each leg is equal to, don't try to add or subtract points from the ordered pairs. Instead, think of how many points you need to move to the right/left or up/down on the x/y axis to get to the new point formed by the right triangle. (pg. 167, geometry book)

If given a list of formulas for lines and asked which one would result in no points being in a certain quadrant on a graph, what should you do first?

You should redraw the graph onto your paper and then draw a line that fits the description of not having points in a certain quadrant. Then, figure out what must be true about the formula for the line you drew. Does it have a negative or positive slope? Is the slope a whole number or a fraction (steep or shallow)? Where does it intercept the y-axis, above or below the origin? The answers to these questions will help you figure out which formula is correct from the list because it will fulfill these requirements.

A line passes through the point (-2,-3) and crosses the y-axis above the origin. How would you determine which slopes from a list of options could be the slope of the line?

You should use the points you are given and the point (0,0) (the coordinates of the origin) to calculate a slope using the formula. Then, reason that the possible slope for this line must be steeper than the slope you calculated because the line cannot actually pass through the origin.

When graphing functions, what is the x-axis used to graph and what is the y-axis used to graph?

You use the x-axis for graphing the input of a function and the y-axis for graphing the output of a function. To graph a function in a coordinate plane, you represent each input x and its corresponding output f(x) as a point (x,y) where y = f(x).

When a parabola is in the form y= ax^2 + bx +c but lacks the term bx (ex. y=x^2- 3), what does that tell you about the graph of the parabola?

b=0. The graph will be centered around the y-axis.

What is the Distance Formula?

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Which direction would the graph of f(x) - c be shifted?

downwards by c units

What is a parabola?

the graph of a quadratic equation of the form y = ax^2 + bx + c where a, b and c are constants and a does not equal 0. The intercepts of the parabola are the solutions to the quadratic equation. If a is positive, the parabola opens upward and the vertex is its lowest point, and if a is negative it opens downward and the vertex is its highest point.

Which direction would the graph of f(x + c) be shifted?

to the left by c units

Which direction would the graph of f(x - c) be shifted?

to the right by c units

Which direction would the graph of f(x) + c (when c is any positive number) be shifted?

upwards by c units