Parabolas/Quadratics
Where does the parabola (y^2) = 3x + 5 open?
To the RIGHT
What are the center and radius of a circle whose equation is (x - 3)^2 + (y + 5)^2 = 64?
Center is (3, -5) and Radius is 8
Standard form of a circle is
(x - h)^2 + (y - k)^2 = r^2 so
(x - 3)^2 + (y + 5)^2 = 64 is also
(x - 3)^2 + (y - (-5))^2 = 64 so (h, k) is (3, -5) and r = 8
What is the solution set of (x + 1)(x - 3) > 0 in set-builder notation?
{x|x<-1 U x>3}
Solution:
x + 1 = 0 OR x - 3 = 0
x = -1 OR x = 3
Since the solution set is for y > 0, the solution set is {x|x<-1 U x>3}.
When completing the square of y = x^2−8x−20, what quantity will be added to and subtracted from the equation
16
Solution:
The coefficient of the middle term is -8 so (-8/2)^2 = 16 is added to and subtracted from the equation (to compensate or balance).
The trajectory of water coming out of a water fountain is modeled by the equation h(t) = -12t^2 + 87t + 11 where h is the height of water in feet and t is the time in seconds. How tall is the water fountain?
11 feet
With respect to the parent function y = x^2, which parabola is vertically stretched, A or B?
A) y = 3(x - 2)^2 + 5
B) y = -3/4(x + 1)^2 - 8
A
Solution:
When |a| > 1, there is vertical stretching.
When 0 < |a| < 1, there is vertical compression.
If a circle whose center is at (-6, 9) has diameter of 12, determine the equation of the circle in center-radius form (standard form).
(x + 6)^2 + (y - 9)^2 = 36
Solution:
Radius is half of the diameter so radius = 12/2 = 6.
Thus, the standard form is
(x + 6)^2 + (y - 9)^2 = 36
What is the solution set of -(5x - 3)(2x + 1) ≤ 0 in interval notation?
(-∞, -1/2] U [3/5, ∞)
Solution:
5x - 3 = 0 OR 2x + 1 = 0
x = 3/5 OR x = -1/2
Since the solution set is for y ≤ 0 and the parabola faces downward, the solution set is (-∞, -1/2] U [3/5, ∞).
What is added to and subtracted from the equation y = 2(x^2) + 3x - 2 when completing the square?
9/16 = 0.5625
Solution:
y = 2(x^2) + 3x - 2
y = 2(x^2 + 3/2x) - 2
Therefore, [(3/2)/2 ]^2 = (3/4)^2 = 9/16 = 0.5625 is added to and subtracted from the equation.
What is the name of our Assistant Principal for Mathematics and World Languages?
Mr. Evan Smith
If the parent function of a parabola is y = x^2, what is the vertex form for a parabola that is neither vertically stretched nor compressed but is shifted 8 units to the left and 3 units up?
y = (x + 8)^2 + 3
Solution:
(h, k) is (-8, 3)
so vertex form is y = (x - (-8)^2 + 3
or y = (x + 8)^2 + 3
A circle has a center at (-1, -10) and a point on its circumference is (3, -13). What is its equation in standard form?
(x + 1)^2 + (y + 10)^2 = 25
Solution:
(x - h)^2 + (y - k)^2 = r^2
(3 - (-1))^2 + (-13 - (-10))^2 = 16 + 9 = 25 = r^2
So the standard form is
(x + 1)^2 + (y + 10)^2 = 25
What is the solution set of the inequality x^2 - 9x + 14 ≥ 0 in interval notation?
(-∞, 2] U [7, ∞)
Solution:
x^2 - 9x + 14 = (x - 7)(x - 2) = 0
x - 7 = 0 OR x - 2 = 0
x = 7 OR x = 2
Since the solution set is for y ≥ 0 and the parabola faces upward, the solution set is (-∞, 2] U [7, ∞).
The general equation of a circle is x^2 + y^2 - 10x + 6y - 60 = 0. When completing the squares, what will be the total number added to and subtracted from the equation?
34
Solution:
(x^2 - 10x) + (y^2 + 6y) - 60 = 0
(-10/2)^2 and (6/2)^2 are added to and subtracted from the equation (to compensate/balance). The total number is 25 + 9 = 34
A standard basketball court is 94 feet by 50 feet, and the center circle is 12 feet in diameter. If the basketball court is put in the first quadrant on the xy-coordinate plane so that the length side is on the x-axis and the width side is on the y-axis, what is the standard form of the center circle?
(x - 47)^2 + (y - 25)^2 = 36
Solution:
94/2 = 47 feet and 50/2 = 25 feet so the center of the Basketball Court Center Circle is (47, 25).
The radius (r) is 12/2 = 6 feet and so r^2 = 36. The standard form then is
(x - 47)^2 + (y - 25)^2 = 36
What are the roots of y = 3(x^2) - 20x - 32?
x = -4/3 and x = 8
Solution:
3(x^2) - 20x - 32 = 0
(3x + 4)(x - 8) = 0
3x + 4 = 0 OR x - 8 = 0
So solutions/roots are x = -4/3 and x = 8
A circle has a center at (-5, 8) and a point on its circumference is (-10, -4). What is its area in terms of Pi (π)?
13π
Solution:
(x - h)^2 + (y - k)^2 = r^2
(-10 - (-5))^2 + (-4 - 8)^2 = 25 + 144 = 169 = r^2
Area = (r^2)π = 169π
What is the solution set of the inequality x^2+11x≥7x+32 in set-builder notation?
{x|x≤-8 U x≥4}
Solution:
x^2 + 11x ≥ 7x + 32
x^2 + 11x - 7x - 32 ≥0
x^2 + 4x - 32 ≥0
(x + 8)(x - 4) ≤ 0
The general equation of a circle is x^2 + y^2 + 4x - 6y - 3 = 0.Find the center and the radius.
Center is (-2, 3) and Radius is 4
Solution:
x^2 + y^2 + 4x - 6y - 3 = 0
(x^2 + 4x) + (y^2 - 6y) = 3
(x^2 + 4x + 4) + (y^2 - 6y + 9) = 3 + 4 + 9
(x + 2)^2 + (y - 3)^3 =16
So the center is (-2, 3) and the radius is 4
The bed of one section of a river can be modeled by the parabolic function y = x^2 - 14x + 52 where y is in feet. If graphed on the xy-plane, one bank of the river is exactly on the y-axis, what is the greatest depth of the river at this parabolic section?
Greatest Depth at the section = 49 feet
Solution:
y = (x^2 - 14x) + 52
y = (x^2 - 14x +(-14/2)^2) + 52 - (-14/2)^2
y = (x^2 - 14x + 49) + 52 - 49
y = (x - 7)^2 + 3
Since the vertex is (7, 3) and y-intercept is 52, the greatest depth at the section is 52 - 3 = 49 feet
If the parent function of a parabola is y = x^2, what is the standard form for a parabola that is facing down, vertically compressed so that every y-coordinate is just a third of the y-coordinate of the parent function, and then shifted 4 units to the left and 23 units up?
y = -(x^2) - 8x + 53
Solution:
(h, k) is (-4, 23), the parabola faces down, and the vertical compression factor is 1/3
so vertex form is y = -1/3(x +4)^2 + 23
y = -1/3(x^2 + 8x + 16) + 23
y = -1/3(x^2) - (8/3)x - (16/3) + 23
y = -1/3(x^2) - (8/3)x + 53/3
y = -(x^2) - 8x + 53
A circle whose radius is 15 has a center at (7, 8) and a point on its circumference is (16, y). Find y.
y = 20
Solution:
(x - h)^2 + (y - k)^2 = r^2
(16 - 7)^2 + (y - 8)^2 = 15^2
81 + (y^2 - 16y + 64) = 225
y^2 - 16y + 145 - 225 = 0
y^2 - 16y - 80 = 0
(y + 4)(y - 20) = 0
y + 4 = 0 OR y - 20 = 0
Solutions are y = -4 AND y = 20
Choose y = 20 because it is the positive value (makes sense).
What is the solution set of the inequality -2(x^2)-5x+50 ≤ -3x-10 in interval notation?
(-∞, -6] U [5, ∞)
Solution:
-2(x^2)-5x+50 + 3x + 10 ≤0
-2(x^2)-2x+60 ≤0
-2(x^2+ x - 30)
-2(x + 6)(x - 5) ≤ 0
The general equation of a circle is x^2 + y^2 + 6x − 4y - 12 = 0. What is its area in terms of Pi (π)?
25π
A circular lawn can be modeled by the general equation of a circle that is x^2 + y^2 + 6x + - 12y - 55 = 0 where the garden dimensions are in feet. If the lawn owner pays $0.75 each square foot for grass cutting, how much does the owner pay for grass-cutting for the entire circular lawn? Use pi = 3.14 and round your answers to the nearest cent.
Solution:
(x^2 + 6x) + (y^2 - 12y) = 55
(x^2 + 6x + 9) + (y^2 - 12y + 36) = 55 + 9 + 36
(x + 3)^2 + (y - 6)^2 = 100 so radius is 10 feet
Area of circular lawn = (3.14)(100) = 314 square feet
Total grass-cutting cost = 314($0.75) = $235.50