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Mathematics > Tables equations and graphs > Flashcards

Tables equations and graphs Flashcards

1
Q

Find the equation of a linear graph

using y-intercept and gradient

A

y = mx + c
• m is the gradient (rise/run)
• c is the y intercept

y = __
• A horizontal line where __ is the y-intercept

x = __
• A vertical line where __ is the x-intercept

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2
Q

Find the equation of a linear graph

using x-intercept and y-intercept

A

ax + by = c
• a and b are numerical values

  1. Make x = 0 to find the y-intercept, plot
  2. Make y = 0 to find the x-intercept, plot
  3. Join the points with a ruled line and arrows on either end
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3
Q

Writing equations from a linear graph

A
  1. Find the gradient
    (rise/run, work from left to right)
  2. Find the y-intercept
    (if it is not clear, substitute the x, y values of a point the line goes through into the equation y = mx + c and solve to find ‘c’)
  3. Write in the form of y = mx + c
    (dependant on the question, x, y values may be different letters)
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4
Q

Piecewise functions

A
  • A solid circle indicates the point which IS included with its associated line
  • A hollow circle indicates the point which IS NOT included with its associated line
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5
Q

Describing piecewise functions

A

• The y-intercept is at __ for (what graph 1 represents in context to question), but it is at __ for (what graph 2 represents in context to question).

• The ‘steps’ are greater for (graph 1/2). Between __ and __ (x value of graph), the (y value of graph) (drops/rises) __ compared with __ for (graph 1/2). Between __ and __ (x value of graph), the (y value of graph) (drops/rises) __ compared with __ for (graph 1/2).
OR
• Both graphs go up in ‘steps’ of __. The steps for (graph 1) go up by intervals of ____ for each __, whereas the steps for (graph 2) go up by intervales of ____ for each __.

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6
Q

Domain

A

• What are the relevant values of y from x

Eg. y = _ when __ ≤ x ≤ __

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7
Q

Drawing an exponential graph

A
  1. Make a table
    (To find the y value for a negative x value, make the y value into a fraction and remove the negative sign, then solve.

Eg. x = -1, y = 2^-1, = 1/2^1, = 1/2)
2. Plot the points and join with a line to form a smooth curve
3. Draw in asymptote
(Make sure the line does not touch the asymptote)

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8
Q

Exponential graphs

A

The basic exponential graph is y = p^x
• Has asymptote of y = 0
• Passes through 0,1
• When x = 1, the base is the y value

• If the base number is changed, the shape of the curve is changed
(If base > 1 the graph is a growth curve)
(If base < 1 the graph is a decay curve)

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9
Q

Translated exponential graphs

A

y = p^(x±b) ± c

• b value indicates the horizontal movement of the graph
(+ value moves left, - value moves right)
• c value indicates the vertical movement of the graph
(+ value moves up, - value moves down)
• p value indicates the shape of the graph
(base number)
• -p value reflects the points of the graph in the horizontal asymptote

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10
Q

Writing equations from exponential graphs

A
  1. Draw the asymptote
    (if asymptote is at y = 0 and the graph passes through (0,1) this means the exponential is in the basic position and will have the equation of y = p^x).
  2. Mark the point (0,1) and draw arrows from this point to where the graph is 1 unit above the asymptote
  3. Find how many units the point has moved left/right, up/down from and substitute into the equation for b and c
  4. Select a point the graph passes through and substitute into the equation for x and y to find p (use y intercept if given)
  5. Write equation without using x or y values
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11
Q

Parabolas

A

The basic parabola graph is y = x^2
• Passes through 0,0

• If the base number is changed, the shape of the curve is changed
(If 0 < base < 1 the graph opens up wider)
(If base > 1 the graph closes up thinner)
• When a ≠ 1, draw the table and plot the points

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12
Q

Drawing a parabola

A

If the equation is in the form of y = a(x±p)(x±q)
1. Plot the x-intercepts with the opposite p and q values
(Using the opposite charge (positive <> negative)
2. Plot the y-intercept with the a value
(Using y = a(p) x (q) if the equation has an a value, otherwise using y = (p) x (q)
3. Draw an axis of symmetry half way between the x-intercepts
4. Substitute x coordinate of the axis of symmetry into the equation to find the turning points
(Make x = axis of symmetry)
5. Substitute to find extra points using different x values
6. Use the axis of symmetry and reflection to plot missing points

If the equation is in a different form
• Complete a table and plot points

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13
Q

Writing equations from parabolas

Method 1 • x-intercepts

A

Equation is in the form of y = a(x±p)(x±q)

  1. Take the x-intercepts with the opposite p and q values
    (Using the opposite charge (positive <> negative)
  2. Select a point the graph passes through and substitute into the equation for x and y to find a
  3. Write equation without using x or y values
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14
Q

Writing equations from parabolas

Method 2 • turning point

A

Equation is in the form of y = a(x±b)^2 ± c

  1. Find how many units the turning point has moved left/right, up/down from (0,0) and substitute into the equation for b and c
    • b value indicates the horizontal movement of the graph
    (+ value moves left, - value moves right)
    • c value indicates the vertical movement of the graph
    (+ value moves up, - value moves down)
  2. Select a point the graph passes through and substitute into the equation for x and y to find a
    (Use the y-intercept)
  3. Write equation without using x or y values
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15
Q

Translating, describing and comparing parabolas

A

• Give coordinates of important points on the parabolas
(x-intercepts, y-intercept, turning point)
• Compare parabolas using words
(Eg. The graph has been reflected in the line y = 1.
The graph has been translated left 3 units and up 2 units.
The graph is twice as steep/wide.)
• State the values of x where y is positive or negative
(Eg. y is negative if x is between __ and ___,
( __ < x < ___ ).
y is positive if x is less than __ or more than ___,
( x < __ and x > ___ ).

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16
Q

Writing equations for translated parabolas from a given equation

A
  1. Vertical movement
    • To move a parabola up, add the required number of units at the end of the equation
    • To move a parabola up, subtract the required number of units at the end of the equation
  2. Horizontal movement
    (First factorize the equation, if it is not already)
    • To move a parabola left, add the required number of units to each of the brackets in the equation
    • To move a parabola right, subtract the required number of units from each of the brackets in the equation

Eg. Translate the graph of y = (x - 2)(x + 4) to the right 3 units and up 5 units.

  1. Equation becomes y = (x - 2)(x + 4) + 5
  2. Equation becomes y = (x - 5)(x + 1) + 5

Eg. Translate the graph of y = x^2 - 2x to the left 4 units and down 3 units.

(Equation becomes y = x(x - 2)

  1. Equation becomes y = x(x - 2) - 3
  2. Equation becomes y = (x + 4)(x +2) - 3
17
Q

How to find the nth term of a quadratic sequence

A

y = ax^2 + bx + c

  • 2a = second difference
  • 3a + b = first difference
  • a + b + c = first y value

(all for the first x = 1 value in the table)

  1. Solve equations in order to find a, b and c
  2. Write out the equation in the form y = ax^2 + bx + c

Eg.

x       y         differences
1        1           > 6   > 6
2       7          > 12  > 6
3       19         > 18  
4       37        > 24
5       61
2a = 6,
a = 3

3a + b = 6,
3(3) + b = 6,
9 + b = 6,
b = -3

a + b + c = 1,
3 - 3 + c = 1,
0 + c = 1,
c = 1

• 3n^2 - 3n + 1

18
Q

Parabolic equation

A

y = p^(x±_) x ____

p • What the pattern is increasing/decreasing by
If increasing the number will be > 1 and will be calculated by adding the % change to 100
If decreasing the number will be < 1 and will be calculated by subtracting the % change from 100

Eg. Pattern increases by 25%
100 + 25 = 125,
p = 1.25

Eg. Pattern decreases by 25%
100 - 25 = 75,
p = 0.75

_ • If there is a horizontal movement of the graph, this is the number which will indicate it
(+ value moves left, - value moves right)

____ • What y equals when x is 0
(“day one” value)

Mathematics (12 decks)

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