Completing the Square
Identify the vertex from the following quadratic function in vertex form:
f(x): -(x-2)2+12
Answer:
f(x): -(x-2)2+12
Vertex: (2,12)
One day Humza decides to ride the mini xcelerator at Knott's Berry Farm. His cart takes off and rises 18 feet in the air. Use the equation h(x)=-2x2 +4x+16 to show the height of Humza's cart after x seconds.
Standard Form:h(x)=-2x2 +4x+16
Factored Form:h(x)= -2(x+2)(x-4)
Vertex Form: h(x)= -2(x - 1)2 +18
1) Find the inputs (x-values) and outputs (y-values) based on the information given from the word problem.
Answer: See graph A for more information.
Fill in the blanks and complete the square. Write in vertex form and include vertex:
f(x)= 2x2-8x+5
Step1: f(x)-5=2x2-8x
Step 2: f(x)/2 -(5/2)= x2-4x b/2=__ /2= (__)2=_
Step 3: f(x)/2 -(5/2)= x2-4x+4
Step 4: f(x)/2 -(5/2) = (x-__)2
Step 5: f(x)/2 = (x-__)2- 3/2
Step 6: 2*[f(x)/2] = [(x-2)2- 3/2]*2
Step7: f(x)= __(x-__)2-__
vertex: ( , )
Answer:
f(x)= 2x2-8x+5
Step1: f(x)-5=2x2-8x
Step 2: f(x)/2 -(5/2)= x2-4x b/2= 4/2=(2)2= 2
Step 3: f(x)/2 -(5/2)= x2-4x+4
Step 4: f(x)/2 -(5/2) = (x-2)2
Step 5: f(x)/2 = (x-2)2- 3/2
Step 6: 2*[f(x)/2] = [(x-2)2- 3/2]*2
Step7: f(x)= 2(x-2)2-3
vertex: (2,-3)
Identify the zeros from the following quadratic function that;s in factored form:
f(x)= (x+4)(x-2)
Answer:
f(x)= (x+4)(x-2)
x+4=0
x-2=0
zeros: x=-4 ,x=2
Create an equation that fits the transformations:
Translate 8 to the left
Translate 3/2 down
Answer: f(x)= (x+8)2 -3/2
One day Humza decides to ride the mini xcelerator at Knott's Berry Farm. His cart takes off and rises 18 feet in the air. Use the equation h(x)=-2x2 +4x+16 to show the height of Humza's cart after x seconds.
Standard Form:h(x)=-2x2 +4x+16
Factored Form:h(x)= -2(x+2)(x-4)
Vertex Form: h(x)= -2(x - 1)2 +18
2) What is the height of Humza's cart at the time that it is nearly at the top?
Answer:
Humza's cart is 16 feet that's nearly at the top.
Convert into vertex form by completing the square.Make sure to include vertex in your final answer.
f(x)=x2+6x-7
Answer: f(x) (x+3)2-16
v(-3,-16)
Identify the vertex from the following quadratic function:
f(x)= x2-6x+5
Answer:
f(x)= x2-6x+5
-b/2a = -(-6)/2(1)= 3
f(x)= (3)2-6(3)+ 5= -4
vertex: (3,-4)
Create an equation that fits the transformations:
-Vertical stretch by 3
-Reflect across the x-axis
-Translate 6 to the right
-translate 3 up
Answer: f(x)= -3(x-6)2 +3
One day Humza decides to ride the mini xcelerator at Knott's Berry Farm. His cart takes off and rises 18 feet in the air. Use the equation h(x)=-2x2 +4x+16 to show the height of Humza's cart after x seconds.
Standard Form:h(x)=-2x2 +4x+16
Factored Form:h(x)= -2(x + 2)(x - 4)
Vertex Form: h(x)= -2(x - 1)2 +18
2) When does Humza's cart reach its max height?
Answer: Humza's cart reaches its max height at 1 second.
Convert into vertex form by completing the square.Make sure to include vertex in your final answer.
f(x)= -3x2-12x+1
Answer:
f(x)= -3(x+2)2+13
vertex(-2,13)
Graph the following equation in factored form:
g(x)= (x-4)(x+4)
Answer:
g(x)= (x-4)(x+4)
x-values: -4, -1, 0, 1, 4
y-values: 0, -15, -16, -15, 0
Write the sequence of transformations for the following equation:
f(x)= (x-3)2-2
Answer:
- shift 3 right
-shift 2 down
Emily stands on a balcony and throws a paper airplane to Alexis who's at ground level. The airplane's height (in feet above the ground), x seconds after Emily threw it, is modeled by the following:
g(x)=- x2 + 4x + 3
4) At what time(s) will the airplane be at least 3 feet tall.
Answer: The airplane will be at least 3 feet tall at 0 and at 4 seconds.
0<x<4
Write the equations of the given tables in both vertex and standard form.
X-values: -4, -3, -2, -1, 0
Y-values: 0, 3, 4, 3, 0
Standard Form: f(x) =-x2-4x
Vertex Form: f(x)= -(x + 2)2 + 4
Convert into vertex form by completing the square.Make sure to include vertex in your final answer.
f(x)= 5x2+10x-4
Answer:
f(x)= 5(x+1)2 -9
vertex: (-1,-9)
Graph the following equation in vertex form:
h(x)= -3(x-2)2+5
Answer:
h(x)= -3(x-2)2+5
x-values: 4,3,2,1,0
y-values:-7,2,5,2,-7
Write the sequence of transformations for the following equation:
w(x)= -3/4(x+8)2 +5
Answer:
-Vertical compression of 3/4
-Reflect over x-axis
-Translate 8 units left
-Translate up 5 units
Emily stands on a balcony and throws a paper airplane to Alexis who's at ground level. The airplane's height (in feet above the ground), x seconds after Emily threw it, is modeled by the following:
g(x)=- x2 - 4x + 3
Answer: The height of the balcony that Emily is throwing the paper airplane to Alexis is 3 feet tall.
(0,3)
Write the equations of the given tables in both vertex and standard form.
X-values: -1, 0, 1, 2, 3
Y-values: 2, -1, -2, -1, 2
Standard Form: f(x) = x2-2x-1
Vertex Form: f(x) = (x-1)2-2