Analytic Geometry Exercices Flashcards
What is a circle?
A set of points that all have equal distance from a fixed point called a center.
What is the equation of a line?
y=mx+b
How do you find the slope of a line?
If the slope of a line is positive then as x ____________, y ______________.
increases
If the slope of a line is negative then as x ____________, y ______________.
x increases y decreases
The MORE the slope is positive or negative, the ___________ the line.
steeper
If the slope is zero, then the line is _______________
horizontal
In the equation ‘‘y=mx+b’’, what is b?
a constant, the y-intercept
What is the the point-slope formula of a line?
y - y1= m (x - x1)
Find the equation of the line L through the points (5, -15) and (50, 10).
How do you express a horizontal line?
y = b, where b is the the y-intercept of the line
How do you express a vertical line?
x = a, where x is the x intercept of the line
Explain why, using 2 points along a line, horizontal lines have a slope of 0.
Horizontal lines have a slope of zero since y2 -y1 = 0 for any two points on the line.
Why do we say that vertical lines are undefined?
Since x2 - x1 = 0, when we plug it in the Dy/Dx formula, anything divided by 0 is undefined.
What are parallel lines?
Two lines that are the same distance apart and have the same slope.
What are perpendicular lines?
Perpendicular lines meet at right angles (90 degrees).
How do you verify that two lines are perpendicular to eachother?
Check that their slopes are negative reciprocals of eachother.
Check that their slopes multiply together to give -1.
Find an equation of the line L2 perpendicular to L1: 2x - 3y = -1 passing through the point (6, -5).
What is the standard form of a given line?
ax + by = c
How could we find the distance D between two points P(x1,y1) and Q(x2,y2)? What would be the formula?
Find the distance between (7, -4) and (3, 2).
Given two points, how do we find the midpoint M between them?
The midpoint M of the line segment connecting (6, -7) and (2, 4) is:
Find the point (x,y) of the line y = -2x that is equidistant from the points A(-3, -4). and B(6, 2).
The equal distance from the center and all the points that make a circle is called the:
radius
Twice the radius is called the:
diameter, it connects opposite points of a circle
Let (h, k) denote the center, r the radius and (x, y) any point on the circle. Find the standard equation of a circle.
The distance from (h, k) and (x, y) is exactly r so:
The standard equation of the distance r between the center and the points of a circle is: (x - 1)² + (y +2)² = 3, what would be the coordinates of the center and the radius?
Center : (1, -2)
Radius: √3
What is the equation of the simplest circle of all circles?
x² + y² = 1 where the center is the origin (0, 0) and a radius of 1.
What is a unit circle?
Any circle with a radius of 1.
Find the equation of a circle that has (-5, -7) and (1, 1) as its endpoint of the diameter and the distance r between its center and a point.
Circle equation: r² = (x + 2)² + (y +3)²
r = 5
Find the standard equation of a circle:
2x² + 12x + 2y² -8y +4 = 0
Identify its radius and its center.
r = √11
Center: (-3, 2)
What is the definition of mathematical modeling?
Adding a numerical structure to a real-world situation
Give an example of mathematical modeling.
Getting in a taxi cab and calculating the total cost of the fare taking into consideration the drop charge and the rate per mile.
What is a linear function?
It is a graph that produces a straight line in the forms of f(x) = b +mx or f(x) = mx + b
f (x) = b + mx is an increasing function if m ___ 0
m > 0
f (x) = b + mx is an decreasing function if m ___ 0
m < 0
Marcus currently owns 200 songs in his iTunes collection. Every month, he adds 15 new songs. Write a formula for the number of songs, N, in his iTunes collection as a function of the number of months, m. How many songs will he own in a year?
1 year = 12 months
N(12) = 200 + 15(12) = 380 songs
N(m) = 200 +15m
The population of the city increased from 23,400 to 27,800 between 2002 and 2006. Find the rate of change of the population during this time span.
m = 1100 per year
The pressure, P, in pounds per square inch (PSI) on a diver depends upon their depth below the water surface, d, in feet, following the equation P(d ) = 14.696 + 0.434d. Interpret the components of this function. (2)
The pressure on the diver increases by 0.434 PSI for each foot their depth increases.
The initial value, 14.696, will have the same units as the output, so this tells us that at a depth of 0 feet, the pressure on the diver will be 14.696 PSI.
If f (x) is a linear function, f(3) = -2, and f(8) = 1, find an equation for the function.
f(x) = -19/5 + 3/5x
Working as an insurance salesperson, Ilya earns a base salary and a commission on each new policy, so Ilya’s weekly income, I, depends on the number of new policies, n, he sells during the week. Last week he sold 3 new policies and earned $760 for the week. The week before, he sold 5 new policies and earned $920. Find an equation for I(n), and interpret the meaning of the components of the equation
I(n) = 520 + 80n
He starts at 520$ and earns 80 for each new policy.
The balance in your college payment account, C, is a function of the number of quarters, q, you attend. Interpret the function C(q) = 20000 – 4000q in words. How many quarters of college can you pay for until this account is empty?
You have an initial balance of 20000 and it decreases at a rate of 4000$ per quarter.
After 5 quarters, your balance will be empty.
Given the table below write a linear equation that represents the table values.
P(w) = 1000 + 40w
Summarize these key concepts:
Definition of Mathematical Modeling
Definition of a linear function Structure of a linear function Increasing & Decreasing functions Finding the vertical intercept (0, b) Finding the slope/rate of change, m Interpreting linear functions
When graphing a linear function, there are three basic ways to graph it: (3)
1) By plotting points (at least 2) and drawing a line through the points
2) Using the initial value (output when x = 0) and rate of change (slope)
3) Using transformations of the identity function f (x) = x
Are the lines below perpendicular, parallel, or neither?
Neither
The equation of the line that goes through the point (4,8) and is parallel to the line 3𝑥+2𝑦=4 can be written in the form 𝑦=𝑚𝑥+𝑏 where 𝑚 _______
b _______
m = -3/2
b = 14
The equation of the line that goes through the point (15,37) and is parallel to the 𝑥-axis can be written in the form 𝑦=𝑚𝑥+𝑏 where
𝑚 _______
b _______
𝑚 = 0
b = 37
An equation of a line through (5, 6) which is perpendicular to the line 𝑦=4𝑥+4 has slope
_____and
y-intercept of ______.
m = -1/4
b = 29/4
For the pairs of lines defined by the following equations indicate with an “I” if they are identical, a “P” if they are distinct but parallel, an “N” (for “normal”) if they are perpendicular, and a “G” (for “general”) if they are neither parallel nor perpendicular.
I
N
G
P
Find an equation of the circle with center at the origin and passing through (−1,6) in the form of (𝑥−𝐴)²+(𝑦−𝐵)²=𝐶 where 𝐴,𝐵,𝐶
are constant. Then:
A =
B =
C =
A = 0
B = 0
C = 37
Find the standard form for the equation of a circle (𝑥−ℎ)² + (𝑦−𝑘)²= 𝑟²
with a diameter that has endpoints of (−7,7)
and (5,3).
h =
k =
r =
h = -1
k = 5
r =2*sqrt(10)
Center (4, 4)
r = 1
Find the center and radius of the circle whose equation is 2𝑥² −3𝑥 + 2𝑦² −8𝑦−1=0
.
Center of circle (0.75, 2)
radius = 2.25
Find the point (𝑥,𝑦) on the line 𝑦=5𝑥−2 that is equidistant from the points (−3,6) and (1,6)
(-1, -7)
A circle has a radius of √37 and is centered at (1.3,−3.5).
Write the equation.
(x−1.3)² +(y+3.5)² =37
