Individual Project
Use the graph of f (x) = x2 to match the function to its corresponding graph. In words describe the transformation that occurs (ex: The graph of f (x) is shifted 6 units to the left).
f (x) = x^
Choose from the following functions:
g (x) = (x -- 2)^2; h (x) = x^2 -- 2; i (x) = (x + 3)^2; j (x) = (x + 1)^2 + 3
a) Answer: i (x)
Function: (x+3)^
Description of transformation: The function is shifted to the left 3 units.
b) Answer: h (x)
Function: x^2-2
Description of transformation: The function is shifted down 2 units.
c) Answer: j (x)
Function: (x+1)^2 +
Description of transformation: The function is shifted to the left 1 unit and up 3 units.
d) Answer: g (x)
Function: (x -- 2)^
Description of transformation: The function is shifted to the right 2 units.
Find the domain of the function and express the answer in interval notation. Explain in words or show the calculations.
a) f (x) = 4x^2 -- 7x +
Answer: x is all real numbers
Show Work or Explain in Words: The domain of a function is defined where all values of x is a real number. Because there is no instance where the function f (x), which is a quadratic function, becomes undefined, it stands that all values of x is continuous. Thus the domain of f (x) is where x encompasses the value of all real numbers.
b) g (x) = 10 / (x+7)
Answer: x for all values except -7
Show Work or Explain in Words: The function g (x) becomes undefined when x+7 = 0. Consequently, x + 7 = 0 when x = -7. Therefore, the function is continuous for all values of x except x = -7. The domain then stands that x = all real numbers EXCEPT -7.
c) f (x) = sqrt (4x -- 16)
Answer: x is equal to or greater than 4
Show Work or Explain in Words: The function f (x) is never negative, the square root will always be positive. Therefore, the value of x has to be where 4x -- 16 is either equal to or greater than 0. Thus, the domain has to be where x is greater than or equal to 4.
d) g (x) = 2x / (x-3)
Answer: x for all values except 3
Show Work or Explain in Words: The function g (x) becomes undefined when x -- 3 = 0. Consequently, x -- 3 = 0 when x = 3. Therefore, the function is continuous for all values of x except x = 3. The domain then stands that x = all real numbers EXCEPT 3.
e) f (x) = 3x -- 9
Answer: x is all real numbers
Show Work or Explain in Words: There is no instance where the function f (x), a line function, becomes undefined. It stands that all values of x is continuous. Thus the domain of f (x) is where x encompasses the value of all real numbers.
3) Find the specified asymptotes of the following functions. Recall that asymptotes are lines therefore the answer must be given as an equation of a line.
a) Find the equation of the vertical asymptote of the function f (x) = 4 / (x+5)
Answer: x = -5
Show Work or Explain in Words: The vertical asymptote of the function f (x) is where the function is undefined. Therefore, x + 5 = 0, where x = -5.
b) Find the equation of the horizontal asymptote of the function g (x) = (5x^2 -- 4) / (x+1)
Answer: There are no horizontal asymptotes.
Show Work or Explain in Words: For the equation g (x), because the numerator has a bigger exponent than the denominator, the function will display a slant asymptote, not a horizontal one.
c) Find the equations of both the vertical and horizontal asymptotes of the function f (x) = (3x -- 1) / (x+4)
Answer: x = -4 and y = 3
Vertical: x = -4
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