Forms of Quadratic Equations
Create a quadratic function that has a vertex at (3, 5) and goes through the point (6, 11).
y = 2/3(x - 3)^2 + 5
Create an equation of a circle that has a center at (3,-1) and goes through the point (6,-1)
(x - 3)^2 + (y + 1)^2 = 9
Solve:
3x^2 - 4 = -9
x = +-(isqrt15)/3
Solve the system:
x^2 + y^2 - 32x + 2y + 7 = 0
3x - y = -1
(1, 4)
You are trying to dunk a basketball and need to jump 2.5 feet into the air to dunk the ball. The height of your feet above the ground over t seconds is given by the function below. Will you be able to dunk the basketball? Explain how you know.
h(t) = -16t^2 + 12t
no (discriminant is negative, or y-value of vertex is 2.25)
Find the vertex of the following equation:
y = -1/3(x - 4)(x + 5)
(-1/2, 27/4)
Find the center and radius of the following circle:
-2x^2 - 2y^2+ 16x - 12y - 10 = 0
Center: (4, -3)
Radius:
2sqrt(5)
Solve:
3x^2 - 2x - 34 = 1 + 6x
x = -7/3, 5
Solve the system:
x^2 + y^2 - 16x + 6y + 36 = 0
x^2 + y^2 - 16x + 12y + 90 = 0
(9, -9), (7, -9)
A toy rocket is launched vertically upward from ground level. The height in feet after t seconds is given by the function below. How long will it take the rocket to hit its maximum height? How long will it take the rocket to hit the ground?
h(t) = -16t^2 + 128t
maximum height in 4 seconds. Return to the ground in 8 seconds.
Create a quadratic function in general form that has zeros at
x = +- 2sqrt(5
and goes through the point (2, 4)
y = -1/4x^2 + 5
Find the vertex, focus, and directrix of the following parabola:
16x - 2y^2 - 8y + 8 = 0
Vertex: (-1, -2)
Focus: (1, -2)
Directrix: x = -3
Solve:
-10/3(x - 5)^2 = 15
x = 5 +- (3isqrt2)/2
Solve the system:
7x^2 + 70x - 9y + 130 = 0
7x^2 + 7y^2 + 70x + 12y + 60 = 0
(-5, -5), (-2, 2), (-8, 2)
Mrs. G.A. has a picture that is 9 inches by 15 inches. She puts it in a picture frame that has a uniform thickness/width. The total area of the picture and the picture frame combined is 216 square inches. What is the thickness/width of the picture frame?
1.5 inches
Rewrite the following equation in vertex form:
y = 2(x + 1)(x - 4)
y = 2(x - 3/2)^2 - 25/2
Create a parabola with the following characteristics:
Vertex: (3, -5)
Directrix:
y = -9/2
(x - 3)^2 = -2(y + 5)
Solve:
-7x^2 - 14x - 1 = -12x
x = (-1 +-isqrt6)/7
Solve the system:
y = 1/4(x - 3)^2
3x - 2y = 13
(5, 1), (7, 4)
Laurent walks along two sides of a rectangular garden bed then across the diagonal to return to where he started. The length of the garden bed is four thirds as long as the width. If Laurent walked a total of 36 meters, what are the dimensions of the garden bed?
9 meters by 12 meters
y = 2x^2 - 8x + 26
Create the equation of the parabola that has the following characteristics:
Focus:
(-11/6, -9/2)
Directrix:
x = -7/6
(y + 9/2)^2 = -4/3(x + 3/2)
Solve:
-5x^2 + 8 = -10x^2 + 8x
x = (4 +-2isqrt(6))/5
Solve the system:
y^2 + 12x + 6y + 20 = 0
-10x^2 + y^2 - 68x + 6y - 130 = 0
(-5, 4), (-5, -10), (-3, 2), (-3, -8)
Mrs. G.A. is selling calculators at the Farmer's Market. When she sets the price at $12 each, she sells 15 calculators per hour. She discovers that for each dollar she lowers the price, she sells 5 more calculators per hour. What is the optimal price to help Mrs. G.A. maximize her revenue per hour?
$7.50