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IB APPLICATIONS > Book 1 Chapter 1 > Flashcards

Book 1 Chapter 1 Flashcards

1
Q

What is the gradient of a line?

A

the slope, the steepness

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2
Q

How do a lot of people think about the gradient (the slope)?

A

rise/run… or change in y / change in x

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3
Q

What is a solution to an equation?

A

A solution is the set of values that makes an equation TRUE.

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4
Q

What does is mean to “satisfy” an equation?

A

Make the equation true.

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5
Q

When you graph a line, what are you actually graphing?

A

THE SOLUTION! Those are all of the points that make the equation true. All of the points that “work”. If you put any other points, (x,y) into the equation, they won’t work, You will get a false statement.

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6
Q

What are the gradients of the diagonals of a square?

A

+1 going up L to R, and -1 going down from L to R

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7
Q

How can you think of a slope of 1 or -1?

A

A perfect diagonal of a square, 45 degrees!

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8
Q

What is an interesting fact about slopes between -1 and +1?

A

HALF OF ALL POSSIBLE SLOPES ARE BETWEEN -1 and +1

(and they are perpendicular to eachother)

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9
Q

WHat is the gradient of a flat surface?

A

ZERO

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10
Q

What is the gradient of a horizontal line?

A

ZERO

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11
Q

What is the gradient of a vertical line?

A

undefined

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12
Q

what is the slope of a line that goes straight up and down?

A

undefined

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13
Q

the number 0/5 is equal to :

A

zero. 0

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14
Q

the number 5/0 is equal to :

A

undefined

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15
Q

Why can’t you divide by zero? (why can zero be in the top and not the bottom?)

A

It doesn’t make sense when you think deeply about what division asks. Example: 10/2 asks “how many times do you add 2 to itself to get to 10?” the answer is 5. 10/0 asks “how many times do you add 0 to istelf to get to 10?” the answer is there is no answer. undefined 0/2 asks “how many times do you add 2 to itself to get to 0?” the answer is zero. you add none of them and get zero. 0/0 asks “how many times do you add 0 to itself to get 0?” the answer is 0, or 1, or 2, or any number, so also, undefined.

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16
Q

What is the x intercept of a line? (what are the coordinates?)

A

Where the line crosses the horizontal axis, the x-axis. The coordinates are ( X, 0 )

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17
Q

What is the y intercept of a line? (what are the coordinates?)

A

Where the line crosses the vertical axis, the y-axis. The coordinates are ( 0, Y )

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18
Q

How can you find the x intercepts from an equation of a line?

A

Plug zero in for Y, and solve for X. The coordinates will be ( X, 0)

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19
Q

How can you find the y intercepts from an equation of a line?

A

Plug zero in for X and solve for Y. he coordinates wll be (0, Y)

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20
Q

what are the coordinates of a point?

A

(x,y) the x and y value.

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21
Q

What is the y coordinate for all x-intercepts?

A

zero. all x-intercepts have a height of zero because they are on the x-axis.

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22
Q

What is the x coordinate for all y-intercepts?

A

zero. All y-intercepts are in the middle, not left or right because they are on the y-axis.

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23
Q

Parallel lines have ______ gradients, or slopes

A

the same

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24
Q

Perpendicular lines have _____ gradients, or slopes

A

OPPOSITE SIGN and the RECIPROCAL

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25
Q

If a line has a slope of 2/3 then any line parallel to it will have a slope of ____ and any line perpendicular will have a slope of ___

A

+2/3 and perp would be -3/2

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26
Q

If a line has a slope of -5, then any line parallel to it will have a slope of __ and any line perpendicular to it will have a slope of ___-

A

-5 and perp would be +1/5

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27
Q

What is the gradient formula? (the slope formula?)

A

m = ( y - y ) / ( x - x )

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28
Q

The three forms of equations of a line we use are :

A

point-gradient form, gradient-intercept form, general form

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29
Q

Why is it called “point-gradient” form?

A

There is a point and a gradient explicitly shown in the form

30
Q

what is point-gradient form?

A

A rearrangement of the gradient formula: y-y =m (x-x)

31
Q

Why is it called “gradient-intercept” form?

A

There is a gradient and a slope explicitly shown in the formula

32
Q

What is the gradient-intercept form?

A

y = mx + c

33
Q

What is the “general” form?

A

ax + by = d no fractions.

34
Q

What is a slightly different from general form you may be asked to write an equation in?

A

ax + by + d = 0 in this case, just bring everything to the left.

35
Q

What is the slope of a line in general form?

A

-a/b

36
Q

How can you always get rid of fractions in any equation?

A

Multiply every term by the LCD (plop the LCD in the numerators, then simplify)

37
Q

what is the “most popular” form?

A

gradient - intercept

38
Q

How can you tell if the point (5, -3) is a solution to a linear equation?

A

If it “works.” Sub 5 in for the X and -3 in for the Y. Simplify. If the statement is true, then it is a solution. If it is a solution, then it is on the line.

39
Q

How can you tell if the point (5, -3) is on a line if you are given the equation?

A

If it “works.” Sub 5 in for the X and -3 in for the Y. Simplify. If the statement is true, then it is a solution. If it is a solution, then it is on the line.

40
Q

What is the graph of a line showing us?

A

It is showing us all of the points, (x,y), that make the equation true. Every dot on the graphed line will make the equation true, they all work.

41
Q

If we are given an X, like x=5, and need to fine a Y that goes with it, how do we find it?

A

substitute the 5 in for the x in the eqution and solve for y, That y value goes with the 5. The point (5, y) will be on the line.

42
Q

If we are given a Y value, like y-8, and need to find the x that goes with it, how do you do it?

A

substitute the 8 in for the y in the equation and solve for x. That x goes with the 8. the point (x, 8) will be on the line.

43
Q

What is the equation of the x axis?

A

y=0

44
Q

What is the equation of the y axis?

A

x=0

45
Q

What is the equation of a vertical line that goes through the x axis at 7? How can you think of that line to help make sense of it?

A

x = 7. You can think “all of the points on this line have an x value of 7, so you are a solution as long as your x=7.

46
Q

What is the eqution of a horizontal line that goes througyh the y axis at 8? How can you think about it to help you?

A

y = 8. Since y is the height, you can think of it as all of the points that have a height of 8. Also, you can think of it as all of the points that have a y value of 8, so you are a solution if your y value is 8.

47
Q

If an x intercept is 455, what point is on the line?

A

(455, 0)

48
Q

If a y intercept is 990, what point is on the line?

A

(0, 990)

49
Q

If you are given two points and need to find the equation of a line, how do you do it?

A

First, fine the gradient using m=(y-y)/(x-x), then use that gradient and EITHER OF THE POINTS. Put it in point gradient form.

50
Q

What is a midpoint?

A

the point halfway along a segment. right in the MIDDLE

51
Q

On a number line, what is the midpoint between 8 and 12?

A

at the average: 10. 10 is halfway between 8 and 12. add and divide by 2. 8+12=20. 20/2= 10

52
Q

How do you find the midpoint between two given points?

A

the X value is the average of the two x values, and the Y value is the average of the two y values.

53
Q

How do you find the average of two numbers?

A

add them up and divide by 2

54
Q

What is the midpoint of (4, 5) and (6, 15). (think, the average of 4 and 6, and the average of 5 and 15

A

average of 4 and 6 is 5. the average of 5 and 15 is 10. so at (5, 10)

55
Q

What is a perpendicular bisector?

A

the line that goes through the midpoint of a segment, and it is also perpendicular to it.

56
Q

How do you find the equation of a perpendicular bisector?

A

Find the point and the gradient (slope) and put it in point-gradient form. The point is the midpoint (use averages) and the slope is the RECIPROCAL and OPPOSITE of the slope between the two given points.

57
Q

What are simultaneous equations? (system of equations)

A

a group of equations

58
Q

What is the solution to simultaneous equations? (system of equations?)

A

The set of points that satisfies (makes true) all of the equations in the system. In algebra 1, this is just ONE POINT, (x,y)

59
Q

In general, what are we looking for with two simultaneous linear equaitioins?

A

THE INTERSECTION! (X, Y) ! the intersection is the solution, it is the point that makes both equations true, it satisfies both equations.

60
Q

With two simultaneous equations, what are the three options for a solution?

A

No solution, one solution, or an infinite number of solutions.

61
Q

When are there no solutions to simultaneous equations?

A

when the lines are parallel (no overlap)

62
Q

When is there just one solution to simultaneous equations?

A

when they cross (at one point)

63
Q

When is there and infinite number of solutions to simultaneous equations?

A

when they are the same line, then all points on the line satisfy both equations

64
Q

What are the three ways we solve simultaneous equations?

A

Graphing (and looking for intersection), substitution and elimination

65
Q

What if you solve simultaneous equations all three ways, what will you notice?

A

they all give the same answer, the same POINT, (x,y)

66
Q

How do you solve simultaneous equations by graphing?

A

graph both lines and look for the intersection, the solution is the coordinates (x,y)

67
Q

How do you solve simultaneous equations by substitution?

A

isolate a variable. SUBSTITUTE into the other equation, solve, then sub back in to find (x,y)

68
Q

How do you solve simultaneous equations by elimination?

A

organize equations so they line up nice. Get coefficients to be opposites. Add equations and solve. Then sub back in to find (x,y)

69
Q

How could simultaneous equations have exactly two solutions?

A

Imagine a line and a parabola. the line could cross the parabola twice.

70
Q

How could simultaneous quadratic equations have no solutions?

(think about what parabolas look like)

A

They could be two parabolas that curve away from each other and never cross.

IB APPLICATIONS (5 decks)

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