Vertex Form & Translating Parabolas
Vertex Form Problems
Complete the Square & Quadratic Formula
Complete the Square & Quadratic Formula Problems
Applied Quadratic Functions Problems
1
How would the parabola move according to the equation:
x2 + c
(Hint: what would happen if c was positive? negative?)
If c is positive, it would move up by c units
If c is negative, it would move down by c units
1
Express the function in vertex form:
a = -2, h = -1, k = -5
What is the vertex?
f(x) = -2(x + 1)2 - 5
vertex: (-1, -5)
1
Write out the quadratic formula. When can you use it?
To find the x-intercepts of a quadratic function in standard form
1
In the function f(x) = 4(x2 − 8x + ____) + 12, what number belongs in the blank to complete the square?
16
1
A juggler is performing her act using several balls. She throws the balls up at an initial height of 5 feet, with a speed of 13 feet per second. If the juggler doesn't catch one of the balls, about how long does it take the ball to hit the floor?
Solve for x: x = -0.285, 1.097
Time can't be negative, so about 1.097 seconds
2
How would the parabola move according to the equation:
(x-h)2
(Hint: what would happen if h was positive? negative?)
If h is negative, it would to the left by h units
If h is positive, it would move to the right by h units
2
Functions f(x) and g(x) are shown:
f(x) = x2
g(x) = x2 - 8x + 21
In which direction and by how many units should f(x) be shifted to obtain g(x)?
Right 4 units, Up 5 units
2
What are two ways to go from standard form to vertex form?
1. Complete the square
2. Find a and the vertex
2
In the function f(x) = 4(x2 − 8x + 16) ____ + 12, what number belongs in the blank to complete the square? (hint: check signs)
-64
2
A golf ball is hit from the ground with an initial velocity of 196 feet per second. Assume the starting height of the ball is 0 feet. How long will it take the golf ball to hit the ground?
Find the solutions: x = 0, 12.25
We already know the starting height is 0, so take the other solution: 12.25 seconds
3
What information can you get from the vertex form:
f(x) = a (x - h)2 + k
(Hint: 4 parts)
a = concavity
h = moving left/right
k = moving up/down
(h, k) = vertex
3
Functions f(x) and g(x) are shown:
f(x) = x2
g(x) = x2 + 18x + 15
In which direction and by how many units should f(x) be shifted to obtain g(x)?
(x+9)2 - 66
Left 9 units, Down 66 units
3
What is the discriminant/radicand? What does it tell you?
The expression under the square root of the quadratic formula
It tells you how many x-intercepts the quadratic function has
3
What is the standard form of 5(x - 2)2 - 5?
Solve for its solutions by using the quadratic formula.
5x2 - 20x + 15
x = 1, 3
3
Jules kicks a soccer ball off the ground and into the air with an initial velocity of 25 feet per second. Assume the starting height of the ball is 0 feet. What is the maximum height that the ball reaches?
Find the vertex: (0.781, 9.766)
Max height: 9.766
4
What is the standard form of an equation for projectile motion?
½ at2 + vt + x
4
Convert to vertex form:
4x2 + 12x - 3
Then find the concavity, vertex, y-intercepts, x-intercepts
4(x + 1.5)2 - 12
Concavity: up
Vertex: (-1.5, -12)
y-intercept: (0, -3)
x-intercepts: (0.232, 0), (-3.232, 0)
4
How can you tell how many x-intercepts a quadratic function has?
b2 - 4ac > 0 => 2 x-intercepts
b2 - 4ac = 0 => 1 x-intercept
b2 - 4ac < 0 => nox-intercepts
4
The cost to produce a product is modeled by the function f(x) = 8x2 − 32x + 48 where x is the number of products produced. Complete the square to determine the minimum cost of producing this product.
8(x - 2)2 + 16
$16
4
A particle is moving along a projectile path where the initial height is 160 feet with an initial speed of 144 feet per second. What is the maximum height of the particle?
Find vertex: (4.5, 484)
Max height: 484 feet
5
What is the domain and range for quadratic functions? (Hint: How is concavity related to the range?)
Domain: All real numbers
Range: y-coordinate of the vertex, depending on the concavity
If the parabola is going up, it will be y >= y-coordinate
If the parabola is going down, it will be y <= y-coordinate
5
Convert to vertex form:
3x2 + 12x - 10
Then find the concavity, vertex, y-intercepts, x-intercepts
Concave up
Vertex: (-2, -22)
y-intercept: (0, -10)
x-intercepts: (-4.708, 0), (0.708, 0)
5
What is the process for completing the square for the standard form of a quadratic function?
(hint: use 2x2 + 8x - 20 as an example)
1. Factor out a if it's not 1 from the terms with x
ex: 2(x2 + 4x) - 20
2. Identify b (the term with the x)
b = 4
3. Add (b/2)2 inside the parenthesis, subtract a(b/2)2 outside the parenthesis
2(x2 + 4x + 4) - 8 - 20
4. Simplify
2(x + 2)2 - 28
5
Studies show that employees on an assembly line become more efficient as their level of training goes up. In one company, the number of products P, produced per day, at a level of training of t hours, follows the quadratic model:
P = -0.5t2 + 10t - 8
Complete the square to determine the hours of training that will give maximum products.
Find vertex: (10, 42)
10 hours for 42 products
5
Alexei wants to hang a mirror in his boat, and put a frame around it. The mirror and frame must have an area of 19 square feet. The mirror is 2 feet wide and 6 feet long.
Which quadratic equation can be used to determine the thickness of the frame, x?
What is the maximum thickness, x, that the frame can be?
4x2 + 16x
Solve for solutions: x = -4.958, 0.958
Length cannot be negative, so x = 0.958