algebra

Intermediate Algebra 1033C Page (1)

QUIZ 2A: Quadratic Functions and Equations

Student Name: Last, First Valencia ID:

1) Solve: 𝒙𝟐 − 𝒙 = 𝟓𝟔

SOLUTION:

2) Solve: 𝟐𝒙𝟐 − 𝟏𝟖𝒙 = 𝟎

SOLUTION:

3) Solve: 𝟔𝒙𝟐 + 𝟔 − 𝟏𝟑𝒙 = 𝟎

SOLUTION:

4) Solve: 𝟒𝒙𝟐 − 𝟐𝟓 = 𝟎

SOLUTION:

5) Solve: −𝟐𝒙𝟑 + 𝟏𝟐𝒙𝟐 = 𝟏𝟔𝒙

SOLUTION:

6) Solve: 𝟗𝒙𝟐 + 𝟐𝟓 = 𝟑𝟎𝒙

SOLUTION:

7) Solve: (𝟕 − 𝟐𝒙)𝟐 − 𝟏𝟖 = 𝟒𝟖

SOLUTION:

Intermediate Algebra 1033C Page (2)

8) Solve: (𝟐𝒙 − 𝟓)𝟐 + 𝟐𝟕 = 𝟏𝟏

SOLUTION:

9) Solve: 2𝒙𝟐 − 𝟒𝒙 = 𝟑

SOLUTION:

10) Solve: 3𝒙𝟐 + 𝟒𝒙 + 𝟓 = 𝟎

SOLUTION:

11) Solve: 𝟒 − 𝟏

𝒙 =

𝟑

𝒙𝟐

SOLUTION:

12) Solve: 𝒙 + 𝟏𝟐

𝒙 = 𝟕

SOLUTION:

13) Solve: 𝟐(𝟐𝒙 − 𝟑)𝟐 − 𝟓(𝟐𝒙 − 𝟑) = 𝟑 SOLUTION:

14) Solve: 𝟑𝒙

𝒙+𝟐 +

𝟏

𝒙−𝟏 =

𝟒−𝟕𝒙

𝒙𝟐+𝒙−𝟐

SOLUTION:

Intermediate Algebra 1033C Page (3)

15) Solve: 𝟐𝒙

𝒙−𝟑 +

𝟏

𝒙 = 𝟒

SOLUTION:

16) Find the missing term/terms

i) … + ⋯ + 𝟐𝟓 = (𝟐𝒙 + ⋯ )𝟐 ii) 𝟐𝟓𝒙𝟐 − ⋯ + 𝟏 = (… … … )𝟐

iii) … − 𝟔𝒙 + 𝟗 = (… … … )𝟐 iv) … − 𝟐𝟖𝒙 + ⋯ = (𝟐𝒙 … … )𝟐

v) … − 𝟔𝟔𝒙 + ⋯ = (… … 𝟏𝟏)𝟐 vi) … + ⋯ + 𝟗 = (𝟒𝒙 + ⋯ )𝟐

vii) … + 𝟏𝟎𝟒𝒙 + ⋯ = (𝟐𝒙 + ⋯ )𝟐 viii) … − 𝟕𝟎𝒙 + 𝟐𝟓 = (… … … )𝟐

17) Solve: 𝒙 = 𝟑√𝒙 SOLUTION:

18) Solve: √𝒙 − 𝟏 = 𝒙 − 𝟕

SOLUTION:

19) Solve: √𝟐𝒙 − 𝟑 + 𝒙 = 𝟑

SOLUTION:

20) Solve: √𝟏𝟓 − 𝟐𝒙 = 𝒙

SOLUTION:

21) Solve: |𝒙𝟐 + 𝒙| = 𝟏𝟐

SOLUTION:

Intermediate Algebra 1033C Page (4)

22) Find two consecutive whole numbers such that the sum of their squares is 𝟏𝟏𝟑. SOLUTION:

23) Find the number 𝒃 for which 𝒙 = 𝟐 is a solution of the equation 𝒙 + 𝟐𝒃 = 𝒙 − 𝟒 + 𝟐𝒃𝒙 SOLUTION:

24) The equation 𝟑𝒙𝟐 + 𝟐𝒙 + 𝒌 = 𝟎 has repeated root. Find the value of 𝒌. SOLUTION:

25) Find 𝒌 such that the equation −𝟑 + 𝒌𝒙 − 𝟐𝒙𝟐 = 𝟎 has repeated root. SOLUTION:

26) Find 𝒌 such that the equation 𝒌𝒙𝟐 + 𝒙 + 𝒌 = 𝟎 has a repeated real solution. SOLUTION:

27) The function 𝒇 is defined by (𝒙) = 𝒙𝟐 − 𝟑𝒙 + 𝟓 . Find the two values of 𝒙 for which 𝒇(𝒙) = 𝒇(𝟒) SOLUTION:

28) The function 𝒇 is defined by (𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 . Given that 𝒇(𝟎) = 𝟔, 𝒇(−𝟏) = 𝟏𝟓 and 𝒇(𝟏) = 𝟏, find the values of 𝒂, 𝒃 𝒂𝒏𝒅 𝒄. SOLUTION:

Intermediate Algebra 1033C Page (4)

29) If a toy rocket is launched vertically upward from ground level with an initial velocity of

𝟏𝟐𝟖 𝒇𝒆𝒆𝒕 𝒑𝒆𝒓 𝒔𝒆𝒄𝒐𝒏𝒅, then its height 𝒉 after 𝒕 seconds is given by the equation

𝒉(𝒕) = −𝟏𝟔𝒕𝟐 + 𝟏𝟐𝟖𝒕 (if air resistance is neglected).

i) How long will it take for the rocket to return to the ground?

ii) After how many seconds will the rocket be 112 feet above the ground?

iii) How long will it take the rocket to hit its maximum height?

iv) What is the maximum height?

SOLUTION:

30) Joseph jumped off of a cliff into the ocean in Miami while vacationing with some friends. His

height as a function of time is given by 𝒉(𝒕) = −𝟏𝟔𝒕𝟐 + 𝟑𝟐𝒕 + 𝟔𝟐𝟎 , where 𝒕 is the time in seconds

and 𝒉 is the height in feet.

i) How long did it take for Joseph to reach his maximum height? ii) What was the highest point that Joseph reached? iii) Joseph hit the water after how many seconds? SOLUTION:

31) A falcon dives toward a pigeon on the ground. Its height as a function of time is given by (𝒕) = −𝟏𝟔𝒕𝟐 − 𝟐𝟒𝟎𝒕 + 𝟏𝟓𝟐 , where 𝒉 is the height, 𝒕 is the time. Estimate the time the pigeon has to escape. SOLUTION:

Intermediate Algebra 1033C Page (5)

32) A golf ball is hit from the top of a tower. Its height defined by the function

, 𝒉(𝒕) = −𝟒𝒕𝟐 + 𝟑𝟐𝒕 + 𝟖𝟎, where 𝒕 represents the time in seconds since the ball was hit, and 𝒉

represents the height of the ball above the ground in metres.

i) What is the height of the tower?

ii) How long will it take the ball to hit its maximum height?

iii) Find the maximum height reached by the ball?

iv) After how many seconds the ball will hits the ground?

SOLUTION:

33) A baseball is thrown from an initial height of 𝟑 𝒎 and reaches a maximum height of 𝟖 𝒎,

𝟐 𝒔𝒆𝒄𝒐𝒏𝒅𝒔 after it is thrown.

i) Write a quadratic equation which models this situation.

ii) At what time does the ball hit the ground?

SOLUTION:

HINT: Quadratic Vertex Form

𝒚 = 𝒂(𝒙 − 𝑯)𝟐 + 𝑮

34) The area of a rectangular window is to be 𝟏𝟒𝟑 square centimeters. If the length exceeds the

width by 2 centimeter, what are the dimensions?

SOLUTION:

Intermediate Algebra 1033C Page (6)

35) The sum of the consecutive integers 𝟏, 𝟐, 𝟑, . . , 𝒏 is given by the formula 𝟏

𝟐 𝒏(𝒏 + 𝟏) How many

consecutive integers, starting with𝟏, must be added to get a sum of 𝟔𝟔𝟔? SOLUTION:

36) If a polygon of n sides has 𝟏

𝟐 𝒏(𝒏 − 𝟑) diagonals, how many sides will a polygon with 65

diagonals have? Is there a polygon with 80 diagonals? SOLUTION:

37) Let 𝒇(𝒙) = 𝒙𝟐 − 𝒙 + 𝟒 𝐚𝐧𝐝 𝒈(𝒙) = 𝟑𝒙 − 𝟓. Find (𝒇 − 𝒈)(𝟑), 𝐚𝐧𝐝 (𝒈 + 𝒇)(−𝟐) SOLUTION:

38) Let 𝒇(𝒙) = 𝒙𝟐 − 𝒙 + 𝟒 𝐚𝐧𝐝 𝒈(𝒙) = 𝟑𝒙 − 𝟓. Find (𝒇 ∘ 𝒈)(𝒙), 𝒇(𝒈(𝟐)) 𝐚𝐧𝐝 𝒈(𝒇(−𝟏))

SOLUTION:

39) Determine the quadratic function whose graph is given

Intermediate Algebra 1033C Page (7)

40) Graph the quadratic function 41) Graph the quadratic function

𝒚 = 𝒙𝟐 − 𝟒𝒙 − 𝟓 𝑦 = −(𝒙 + 𝟏)𝟐 + 𝟖

and identify all the key features and identify all the key features

SOLUTION: SOLUTION:

i) Axis of Symmetry: i) Axis of Symmetry:

ii) Vertex : ii) Vertex :

iii) Circle one: Minimum or Maximum iii) Circle one: Minimum or Maximum

iv) Circle one: Opens up or Opens Down iv) Circle one: Opens up or Opens Down

v) 𝒀 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕: v) 𝒀 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕:

vi) 𝑿 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕(𝒔): vi) 𝑿 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕(𝒔):

vii) Domain: vii) Domain:

viii) Range: viii) Range:

ix) 𝒚 is Increase on the interval: ix) 𝒚 is Increase on the interval:

x) 𝒚 is Decrease on the interval: x) 𝒚 is Decrease on the interval:

xi) Standard form: xi) Standard form:

xii) Vertex form: xii) Vertex form:

xiii) Intercept form/ Factored Form: xiii) Intercept form/ Factored Form: