MUF0091 – Quiz 2 Name: __________________________
• Answer the following questions by filling in the blanks.
• Give your answers as fractions using the forward slash (e.g. ½ or ¾) if the answer is not an
integer.
• Use negative symbol ( - ) for any negatives.
• Type 𝑅 to represent all real numbers (if needed).
Q1) The curve of 𝑓(𝑥 ) = −(2𝑥 − 5)(2𝑥 + 3) is shown below
𝟑
a) The coordinates of the negative x- intercept for the graph are (− , 𝟎)
𝟐
𝟓
b) The coordinates of the positive x- intercept for the graph are ( , 𝟎)
𝟐
c) The y- intercept is at (𝟎, 𝟏𝟓)
d) The exact x- values where the function f(x) = -12 are:
−𝟏𝟐 = −𝟒𝒙𝟐 + 𝟒𝒙 + 𝟏𝟓
𝟎 = −𝟒𝒙𝟐 + 𝟒𝒙 + 𝟐𝟕
−𝟒±√𝟏𝟔−𝟒(−𝟒)(𝟐𝟕)
𝒙𝟏,𝟐 =
−𝟖
−𝟒±𝟖√𝟕 𝟏
= = ± √𝟕
−𝟖 𝟐
𝟏
𝑥1 = − √𝟕 (negative x value) and
𝟐
𝟏
𝑥2 = + √𝟕 (positive x value).
𝟐
MUF0091 – Quiz 2 Name: __________________________
e) Use set notation to state the domain and range of the function 𝑓(𝑥 ).
Domain: {𝑥: 𝑥 𝜖 R}
Range: {𝑦: 𝑦 ≤ 𝟏𝟔}
f) Use interval notation to state the x- values where the function f(x) is less than or equal to zero.
𝟑 𝟓
𝑥 𝜖 (−∞, − ] ∪ [ , ∞)
𝟐 𝟐
g) The function g is defined as 𝑔: (−2,3) → 𝑅, where 𝑔(𝑥 ) = −𝑓(𝑥 ).
𝟏 𝟐
𝒈(𝒙) = − [−𝟒 (𝒙 − ) + 𝟏𝟔]
𝟐
𝟏 𝟏
= 𝟒(𝒙 − )𝟐 − 𝟏𝟔 ∴ 𝑻𝑷 ( , −𝟏𝟔)
𝟐 𝟐
Restricted Domain of (-2,3):
𝟏
𝒈(−𝟐) = 𝟒(−𝟐 − )𝟐 − 𝟏𝟔 = 𝟗 → (−𝟐, 𝟗)
𝟐
𝟏
𝒈(𝟑) = 𝟒(𝟑 − )𝟐 − 𝟏𝟔 = 𝟗 → (𝟑, 𝟗)
𝟐
PS: These two points become the endpoints of the graph.
The range of 𝑔 in interval notation is 𝒈(𝒙)𝝐 [−𝟏𝟔, 𝟗)
MUF0091 – Quiz 2 Name: __________________________
• Answer the following questions by filling in the blanks.
• Give your answers as fractions using the forward slash (e.g. ½, ¾) if the answer is not an integer.
• Use the negative symbol ( - ) for any negatives.
Q2) The following questions relate to the function 𝑓: [−2, ∞) → 𝑅, 𝑓(𝑥 ) = 3𝑥 3 + 2𝑥 2 − 7𝑥 + 2
a) If (𝑥 + 2) is a factor, this implies that 𝑓(−𝟐 ) = 𝟎
b) Divide 𝑓(𝑥) by (𝑥 + 2). The quadratic factor of 𝑓(𝑥) is (𝟑𝒙𝟐 − 𝟒𝒙 + 𝟏)
3x2 - 4x + 1
(x+2)√𝟑𝒙𝟑 + 𝟐𝒙𝟐 − 𝟕𝒙 + 𝟐
3x3 + 6x2 -
2
-4x - 7x
-4x2 - 8x -
x + 2
x + 2 -
0
c) Hence, the polynomial 𝑓(𝑥)is fully factorized into: 𝑓(𝑥 ) = (𝑥 + 2) (𝒙 − 𝟏)(𝟑𝒙 − 𝟏)
d) Use your graphic calculator to find the range of 𝑓(𝑥 ), rounding any decimal approximations to two
decimal places.
The range is 𝑓(𝑥 ) 𝜀 [−𝟎. 𝟖𝟗, ∞)
MUF0091 – Quiz 2 Name: __________________________
Q3) The function 𝑓 has an inflection point at (-1,2) and y- intercept at (0, 1).
a) Find the equation of 𝑓 (𝑥 ).
𝒇(𝒙) = 𝒂(𝒙 + 𝟏)𝟑 + 𝟐
(𝟎, 𝟏) → 𝟏 = 𝒂(𝟏)𝟑 + 𝟐
𝒂 = −𝟏
∴ 𝒇(𝒙) = −(𝒙 + 𝟏)𝟑 + 𝟐
b) The graph of 𝑓(𝑥 ) is reflected in y axis, dilated by the factor of 2 from the x- axis, translated 1 unit
up. Find the new equation of 𝑓(𝑥 ).
𝑻𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎𝒂𝒕𝒊𝒐𝒏𝒔: 𝟐𝒇(−𝒙) + 𝟏
𝒇(𝒙) = 𝟐 [−(−𝒙 + 𝟏)𝟑 + 𝟐] + 𝟏
= 𝟐[−(−(𝒙 − 𝟏))𝟑 + 𝟐] + 𝟏
∴ 𝒇(𝒙) = 𝟐(𝒙 − 𝟏)𝟑 + 𝟓
Sketch the graph of the transformed equation and label all important points in exact values.
y
𝒇(𝒙) = 𝟐(𝒙 − 𝟏)𝟑 + 𝟓
𝟑 𝟓
(𝟏 − √ , 𝟎)
𝟐
x
