The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x): Part A: Describe two types of transformations that can be used to transform f(x) to g(x). (2 points)

Part B: Solve for k in each type of transformation. (4 points)

Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x). (4 points)

  A) horizontal translation; vertical translation

  B) H: k = -8; V: k = 16

  C: H: g(x) = 2(x +8) -10; V: g(x) = (2x -10) +16; both ⇒ g(x) = 2x +6

Step-by-step explanation:

A linear function can be translated to another linear function by translating it horizontally, vertically, or some combination of those.

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Point (x, y) on the graph of f(x) will be translated to point (x+h, y+k) to become a point on the graph of g(x). 'h' is the amount of translation to the right; 'k' is the amount of translation up.

horizontal translation: (x, y) = (5, 0) on f(x) moves to (x +k, y) = (-3, 0) on g(x).

  k = -3 -5 = -8 . . . horizontal translation of f(x)

vertical translation: (x, y) = (0, -10) on f(x) moves to (x, y+k) = (0, 6) on g(x).

  k = 6 -(-10) = 16 . . . vertical translation of f(x)

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The equation for f(x) has y-intercept -10 and a slope of 2:

  f(x) = 2x -10

Horizontal translation using the formula in Part A gives you ...

  g(x) = f(x+8) = 2(x +8) -10

Vertical translation gives you ...

  g(x) = f(x) +16 = (2x -10) +16

Additional comment

The translated equations simplify to the same thing in both cases:

  g(x) = 2x +6

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

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step-by-step explanation: