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Math 2 Module 8: Circles & Other Conics

8.1 Ready, Set, Go!

Questions 1, 4, 6, 8

This video will show you how to factor a difference of squares: https://www.youtube.com/watch?v=_qyVzH3e1dY

Questions 2, 3, 5, 7, 9

This video will show you how to factor perfect square trinomials: https://www.youtube.com/watch?v=aM1qf2QN8YE

Questions 10 -15
This video will show you how to find the equation for the graph of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Questions 16 - 24

If the three sides make up a right triangle, what theorem do they have to satisfy? This video will show you how to verify if the three numbers could make up the side lengths of a triangle: https://www.youtube.com/watch?v=cUlKXcorng8

8.2 Ready, Set, Go!

Questions 1 - 10

This video will show you the intuition behind completing the square: https://www.youtube.com/watch?v=EBbtoFMJvFc

This video will show you how to complete the square: https://www.youtube.com/watch?v=xGOQYTo9AKY

Questions 11 - 13

This video will show you how to write the equation of a circle from the given graph: https://www.youtube.com/watch?v=GQXUpB2NHvQ

This video will show you examples of how to find the equation of a circle from the given graph: https://www.youtube.com/watch?v=DBzXAVNKuDc

Question 14 - 19

Find the equation of the circle using the formula. Then, expand your equation by distributing and combining like terms.This video will show you how to write the equation of a circle from the given graph: https://www.youtube.com/watch?v=GQXUpB2NHvQ

To convert the equation into expanded form, evaluate the squared binomial, and then combine like terms. This video will show you how to expand the formula of a circle: https://www.youtube.com/watch?v=bwlOYmI-so8

Question 20 - 25

For this problem, substitute the x and y values of the point into the equation. If the equation holds true, then the point is a solution to the equation. This video will show you how to check if a point is a solution to an equation: https://www.youtube.com/watch?v=_tcM5wQot8Q

8.3 Ready, Set, Go!

Questions 1 - 5

Can you create a right triangle in which the segment connecting the two points is the hypotenuse? Try to use the pythagorean theorem to solve for the length of that segment/hypotenuse. This video will show you how to find the distance between two points: https://www.youtube.com/watch?v=pbeJHyYreHY

Questions 6 - 7

Can you create a right triangle in which the segment connecting the two points is the hypotenuse? Try to use the pythagorean theorem to solve for the length of that segment/hypotenuse. This video will show you how to find the distance between two points: https://www.youtube.com/watch?v=pbeJHyYreHY

This video will show you how to find the distance between two points with irrational coordinates: https://www.youtube.com/watch?v=_fk5hIKP-3k

Question 8

First, find the radius of the circle given the circumference. Use the formula for the circumference of a circle and solve for the radius: https://www.youtube.com/watch?v=yWGeFDXgQvg

Then, plug in the radius and the center point into the equation of a circle. This video will show you how to find the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Question 9

First, find the radius of the circle given the area. Use the formula for the area of a circle and solve for the radius: https://www.youtube.com/watch?v=Xjayo1SJkKM

Then, plug in the radius and the center point into the equation of a circle. This video will show you how to find the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Question 10

Find the center of the circle by solving the system of equations. You will get an ordered pair (h,k). Then, find the radius of the circle, which is half of the diameter. This video will show you how to find the point of intersection of two lines: https://www.youtube.com/watch?v=zkNBSyHh7vw

 

Then, plug in the radius and the center point into the equation of a circle. This video will show you how to find the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Question 11

You're told that the circle is tangent to x = -12 and x = -4, which means that the leftmost point of the circle is at x = -12 and the rightmost point is at x = -4. If you draw vertical lines at x = -12 and x = -4, you'll see that a circle between those lines can only be in quadrant 2 or 3, but the problem already tells you it's in quadrant 2, so you know the circle must sit above the x-axis.

The radius of the circle is half of the diameter, and the diameter must be the distance between x = -12 and x = -4. The problem doesn't specify where the center of the circle is, so you can choose any point as long as the circle doesn't cross over the x-axis, because then it's no longer only in the second quadrant.

After you've picked a center point, plug in the radius and the center point into the equation of a circle. This video will show you how to find the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Question 12

The radius of a circle is the distance from the center to any point in the circle. How could you find the distance between the center and the given point? Can you create a right triangle in which the segment connecting the two points is the hypotenuse? Try to use the pythagorean theorem to solve for the length of that segment/hypotenuse. This video will show you how to find the distance between two points: https://www.youtube.com/watch?v=pbeJHyYreHY

Question 13

You're told that the circle is tangent to y = -2 and y = -17, which means that the bottom point of the circle is at y = -17 and the top point is at y = -2. If you draw horizontal lines at y = -2 and y = -17, you'll see that a circle between those lines will be in quadrant 3 or 4, but the problem already tells you that the center lies on the y-axis itself

The radius of the circle is half of the diameter, and the diameter must be the distance between y = -2 and y = -17.

After you've identified the coordinate of the center point, plug in the radius and the center point into the equation of a circle. This video will show you how to find the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Questions 14 - 15

This video will show you how to find the equation of a circle given three points on the circle: https://www.youtube.com/watch?v=IGABpArcry4

Question 16 - 24

This video will show you the intuition behind completing the square: https://www.youtube.com/watch?v=EBbtoFMJvFc

 

This video will show you how to complete the square: https://www.youtube.com/watch?v=xGOQYTo9AKY

8.4 Ready, Set, Go!

Questions 1 - 4

Try to graph the equations by making a table of values. This video will show you how to graph quadratic equations: https://www.youtube.com/watch?v=3a7UbMJpeIM

Questions 5 - 8

Graph the parabola using the definition of the parabola from the focus and the directrix. This video will explain to you the intuition behind the parabola: https://www.youtube.com/watch?v=okXVhDMuGFg

This video will show you how to graph a parabola given the focus and the directrix: https://www.youtube.com/watch?v=JnbTXC2Gm-Y

Questions 9 - 12

This video will walk you through the parabola folding experiment: https://www.youtube.com/watch?v=wtk5q8wGAe0

Questions 13 - 22

Try to separate the terms with x from the terms with y. Try to complete the square of the terms with x and the terms with y separately. This video will show you how to complete the square: https://www.youtube.com/watch?v=xGOQYTo9AKY

This video will show you how to convert the equation of a circle from expanded form to standard form by completing the square: https://www.youtube.com/watch?v=JbEfcJyOd2Y

8.5 Ready, Set, Go!

Questions 1 - 5

Plug in the x and y values of the ordered pair into the equation. If the equation holds true after simplifying, then the ordered pair lies in the parabola: https://www.youtube.com/watch?v=KKMi7W0Qef4

Question 6

Why is it that you could have found the coordinates of the vertex by looking at the given equation? This video will explain to you how to find the vertex of a parabola: https://www.youtube.com/watch?v=IbI-l7mbKO4

Question 7

Use the distance formula or the pythagorean theorem to model the situation. Find the distance of FM and the distance of MQ in terms of the variables shown. Then, set them equal to each other. The video will show you how to use the distance formula: https://www.youtube.com/watch?v=nyZuite17Pc

Derive the definition of the parabola from the distance of a point in the parabola to the focus and to the directrix. This video will explain to you the intuition behind the parabola: https://www.youtube.com/watch?v=okXVhDMuGFg

Question 8

To find the value of p, you need to use your original equation and find the focus and directrix of the parabola. The video will show you how to find the directrix and focus from a quadratic equation: https://www.youtube.com/watch?v=TkVookljel0

Question 9

Describe "p". Hint: In any parabola, "p” is equal to both the distance between the focus and the vertex AND the distance between the vertex and the directrix. The video will show you how to find the directrix and focus from a quadratic equation: https://www.youtube.com/watch?v=TkVookljel0

Question 10

Find coordinates given parabola graph and directrix. Hint: If two points are on the same horizontal axis, they have the same y-coordinate. If two points are on the same vertical axis, they have the same x-coordinate. The video will show you how to find the directrix and focus from a quadratic equation: https://www.youtube.com/watch?v=TkVookljel0

Questions 11 - 14

This video will show you how to find the equation of a parabola from the vertex and the directrix: https://www.youtube.com/watch?v=MpWP096ctic

The minimum or maximum will be the vertex of the parabola. This video will show you how to find the vertex: https://www.youtube.com/watch?v=IbI-l7mbKO4

Questions 15 - 20

The minimum or maximum will be the vertex of the parabola. This video will show you how to find the vertex: https://www.youtube.com/watch?v=IbI-l7mbKO4

 

This video will show you how to determine if the vertex is a minimum or maximum: https://www.youtube.com/watch?v=6i65SaEVm30

8.6 Ready, Set, Go!

Question 1

Try to plug in the center and the radius into the general equation of a circle. This video will show you how to derive the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

 

This video will show you examples of how to find the equation of a circle from the given information: https://www.youtube.com/watch?v=DBzXAVNKuDc

Question 2

The center of the circle is the midpoint of the diameter. How could you find the midpoint of a segment in a coordinate plane? In addition, the radius of a circle is HALF the distance of the diameter. How could you find the distance of the diameter? This video will show you how to find the distance and midpoint of two points: https://www.youtube.com/watch?v=FzXk_9FQups

Then, plug in the radius and the center point into the equation of a circle. This video will show you how to find the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Question 3

The radius of a circle is the distance from the center to any point in the circle. How could you find the distance between the center and the given point? Can you create a right triangle in which the segment connecting the two points is the hypotenuse? Try to use the pythagorean theorem to solve for the length of that segment/hypotenuse. This video will show you how to find the distance between two points: https://www.youtube.com/watch?v=pbeJHyYreHY

Then, plug in the radius and the center point into the equation of a circle. This video will show you how to find the equation of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

Question 4

This video will show you how to find the equation for the graph of a circle: https://www.youtube.com/watch?v=GQXUpB2NHvQ

This video will show you examples of how to find the equation of a circle from the given graph: https://www.youtube.com/watch?v=DBzXAVNKuDc

Questions 5 - 8

Graph the parabola using the definition of the parabola from the focus and the directrix. This video will explain to you the intuition behind the parabola: https://www.youtube.com/watch?v=okXVhDMuGFg

This video will show you how to graph a parabola given the focus and the directrix: https://www.youtube.com/watch?v=JnbTXC2Gm-Y

Questions 9 - 14

This video will show you how to find the vertex, equation of symmetry, domain, and range of quadratic functions: https://www.youtube.com/watch?v=0LgkWWUQQJg

Question 15

This video describes the focus and directrix of parabolas: https://www.youtube.com/watch?v=pe8Dm_FUpdU

This video describes the vertex form of the parabola equation: https://www.youtube.com/watch?v=IbI-l7mbKO4

8.7H Ready, Set, Go!

Questions 1 - 6

Try to separate radicals and the rest of the terms on separate sides of the equation. Then, how can you get rid of the radical? This video will show you multiple examples of how to solve radical equations: https://www.youtube.com/watch?v=0gicD4STzpg

Questions 7 - 12

This video will explain to you what the equation of the ellipse means by deriving the equation: https://www.youtube.com/watch?v=5TQMJ09MLWM

 

This video will show you how to graph an ellipse from the standard equation: https://www.youtube.com/watch?v=EqLqrX8yJ7k

Question 13 - 16 

This video will explain to you what the equation of the ellipse means by deriving the equation: https://www.youtube.com/watch?v=5TQMJ09MLWM

 

Try to plug in the given values into the derived formula. Then this video will show you how to write the equation of an ellipse given the center, the and the major and minor axes: https://www.youtube.com/watch?v=l-47_pPT8pg

Questions 17 - 18

Try to separate the terms with x from the terms with y. Try to complete the square of the terms with x and the terms with y separately. This video will show you how to complete the square: https://youtu.be/1ba4DaKxM94

 

This video will show you how to graph an ellipse using the standard equation of an ellipse: https://www.youtube.com/watch?v=a86uljNQDsI

Questions 19 - 21

This video will show you how to write the equation of a line in point-slope form: https://www.youtube.com/watch?v=KaohEtGU1sw

If you need a refresher on plotting coordinates on a graph, here's a good video for that: https://www.youtube.com/watch?v=s7NKLWXkEEE

Questions 22

What aspect of a line remains the same after a translation? Try setting up the equation of the lines with the same slope of the original lines that pass through the point (12, -9). This video will show you how to write the equation of a line using a point and the slope: https://www.youtube.com/watch?v=REXFV61M37Q

8.8H Ready, Set, Go!

Questions 1 - 9

This video will show you how to determine if an equation is a parabola, ellipse, hyperbola, or circle: https://www.youtube.com/watch?v=7k-8K9Tsuvw

Questions 10 - 17

This video will explain the intuition behind hyperbolas by deriving the equation for hyperbolas: https://www.youtube.com/watch?v=HPRFmu7JsKU

This video will show you how to graph hyperbolas given the standard equation: https://www.youtube.com/watch?v=Z6cwpsDC_5A

Question 18

Try to separate the terms with x from the terms with y. Try to complete the square of the terms with x and the terms with y separately. This video will show you how to complete the square: https://www.youtube.com/watch?v=xdBSmPja2yo

And this video is an example of a line with a negative slope: https://www.youtube.com/watch?v=dBFbFMvEwQg

Questions 19 - 21

Try to separate the terms with x from the terms with y. Try to complete the square of the terms with x and the terms with y separately. This video shows you how to complete the square: https://www.youtube.com/watch?v=xdBSmPja2yo

This video shows you how to find the asymptotes of a hyperbola: https://www.youtube.com/watch?v=zpKJQAQ4SyI

This video shows you how to find the equation, directrix, and focus of a parabola: https://www.youtube.com/watch?v=doVQ6jbmNkg

This video shows you how to determine the equation, center, and major and minor axes: https://www.youtube.com/watch?v=xQSObYT4Oag

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