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A parabola is defined as a collection of points such that the distance to a fixed point (the focus) and a fixed straight line (the directrix) are equal. But it's probably easier to remember it as the U-shaped curved line created when a quadratic is graphed.
How long is this vehicle, 2001 Ford Focus Hatchback? 4178 mm 164.49 in. How wide is the vehicle, 2001 Ford Focus Hatchback? 1702 mm 67.01 in. What is the curb weight, 2001 Ford Focus Hatchback I 1.8 TDCi (115 Hp)? 1279 kg 2819.71 lbs. What is the gross weight, 2001 Ford Focus Hatchback I 1.8 TDCi (115 Hp)? 1692 kg 3730.22 lbs. How much trunk (boot) space, 2001 Ford Focus Hatchback? 5)Focus(-63/8, -7), Directrix: x= -65/8 The distance between focus and directrix is 2p, so 2p = 1/4, p = 1/8 Since the directrix is a vertical line, the axis is horizontal Since the focus is at y = -7, that is the axis The vertex is at (-64/8,-7) = (-8,-7) So the equation is (y+7)^2 = 1/2 (x+8) See the graph and properties at. Focus are a Dutch progressive rock band formed in Amsterdam in 1969 by keyboardist, vocalist, and flutist Thijs van Leer, drummer Hans Cleuver, bassist Martijn Dresden, and guitarist Jan Akkerman. The band have undergone numerous formations in its history; since December 2016, it has comprised van Leer, drummer Pierre van der Linden, guitarist Menno Gootjes, and bassist Udo Pannekeet.
Many real-world objects travel in a parabolic shape. When you shoot a basketball, the path of the ball creates a parabola.
Example
A parabola that is said to open upwards is shaped like a 'U,' while a parabola said to open downwards is shaped like an upside-down U.
A parabola does not have to be vertical, but horizontal parabolas are not functions (they fail the vertical line test). Some functions you see will be horizontal parabolas with restricted domains (). You will learn more about these when you explore conics.
Example
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The vertex of a parabola is the point where the parabola changes direction, and where the graph is most curved. On graphs of quadratics, it is found at the very top or bottom of the quadratic. The vertex is the point of the parabola at the axis of symmetry.
For quadratics in the standard form ax2 + bx + c, the axis of symmetry can be found using the equation x = . To find the y-coordinate of the vertex, find the axis of symmetry and substitute that x-value into the original equation.
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Calculating the focus and directrix
Below is an example of how to calculate the focus and directrix that may provide a better understanding of the mathematical definition of a parabola provided above:
Example
The focus is a point located on the same line as the axis of symmetry, while the directrix is a line perpendicular to the axis of symmetry. For parabolas, the focus is always on the inside of the parabola, and the directrix never touches the parabola. Because the vertex is the same distance from the focus and directrix, the directrix has a location directly opposite of the focus.
For a parabola in the vertex form y = a(x - h)2 + k, the focus is located at (h, k + ) and the directrix is located at y = k - .
Example
For horizontal parabolas, the vertex is x = a(y - k)2 + h, where (h,k) is the vertex. The focus of parabolas in this form have a focus located at (h + , k) and a directrix at x = h - . The axis of symmetry is located at y = k.
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Vertex form of a parabola
The vertex form of a parabola is another form of the quadratic function f(x) = ax2 + bx + c. The vertex form of a parabola is:
f(x) = a(x - h)2 + k
The a in the vertex form of a parabola corresponds to the a in standard form. If a is positive, the parabola will open upwards. If a is negative, the parabola will open downwards.
In vertex form, (h,k) describes the vertex of the parabola and the parabola has a line of symmetry x = h.
Vertex form is very similar to the general expression for function transformations. Vertex form makes it much easier to graph a parabola because it makes it easy to plot the vertex.
Example
f(x) = -(x - 1)2 + 4
From this equation, we can already tell that the vertex of the parabola is at (1,4), and the axis of symmetry is at x = 1. Now all that has to be done is to plug in points around the vertex, then graph.
You can use completing the square to convert a quadratic in standard form into vertex form. You can also convert to vertex form by using the knowledge that the vertex lies on the axis of symmetry.
The x-value of the vertex is x = - , so the y-value is y = f (-). Plug these values in for h and k in the vertex form equation. Then, a is the same in both forms, so simply copy that to the vertex form equation.
Example
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f(x) = x2 - x - 20
a = 1; b = -1; c = -20
x = h = - = - =
f() = ()2 - - 20 = - - 20 = -20
f(x) = (x - )2 - 20
To convert a parabola in vertex form to standard form, expand the equation and simplify.
Example
f(x) = 5(x + 3)2 - 6
f(x) = 5(x2 + 6x + 9) - 6
f(x) = 5x2 + 30x + 45 - 6
f(x) = 5x2 + 30x + 39
New Orange Warning Zones on Staten Island and in Monroe and Onondaga Counties
New Yellow Precautionary Zones in Upper Manhattan and Nassau and Suffolk Counties
Zone Changes Go into Effect Wednesday for Businesses, Thursday for Schools
South Beach Psychiatric Center Will Be Utilized for COVID-19 Hospitalization Overflow on Staten Island
Positive Testing Rate in All Focus Zone Areas is 4.48 Percent; New York State Positivity Outside All Focus Zone Areas is 2.73 Percent
Statewide Positivity Rate is 3.08 Percent
33 COVID-19 Deaths in New York State Yesterday
Governor Andrew M. Cuomo today announced updated COVID-19 micro-cluster focus zones in New York State. Part of Staten Island's Yellow Precautionary Zone will become an Orange Warning Zone. Orange Warning Zones will also be established in Monroe and Onondaga Counties. New Yellow Precautionary Zones will be established in Upper Manhattan and Nassau and Suffolk Counties. Modifications to current focus zones and new focus zones go into effect this Wednesday, November 25 for businesses and Thursday, November 26 for schools.
The Governor also announced that South Beach Psychiatric Center, a new but unopened Office of Mental Health facility on Staten Island, will at the request of the local hospitals be opening as a temporary emergency facility for COVID patients. The facility served as an emergency hospital in the Spring, providing care for COVID patients.
'We need a reality check because these are dangerous times that we're in. The COVID rate, number of deaths, number of hospitalizations, everything we watch all day on TV, is all a function of our actions. There is no predetermined result here. You tell me what New Yorkers do today, I will tell you the infection rate tomorrow. We are in a place now where there is a bad synergy -- a sense of COVID fatigue. Yes, we're only at 3 percent and only Vermont, Maine, and Hawaii are lower than us, but that can change in an instant and we must remember how we got here,' Governor Cuomo said. 'It's the infection rate in your community that matters, and that's why we talk about micro-clusters. Influence your family's behavior and your community's behavior and be a COVID change agent in your community. We're social beings and it's also the high season of social activity. That is a bad combination, and it is always the combination of events that creates the major issues. Now, Upper Manhattan, basically the Washington Heights area, will become a yellow zone. We're also adding additional yellow zones on Long Island. Parts of Onondaga and Monroe Counties will become orange zones. Staten Island, part is a yellow zone, part is an orange zone. This year, let's think of Thanksgiving as a time to yes, really give thanks to the people who really did phenomenal things this year, and continue to thank them by doing our part and wearing masks, staying socially distant, and practicing safe behaviors.'
Modifications to Current Focus Zones
Staten Island - Parts of Staten Island Move to Orange Warning Zone - Click Here for Map
Parts of Staten Island will transition to an orange warning zone based on increasing 7-day positivity rates, number of cases, and hospitalizations that meet the metrics for orange warning zone designation. The remainder of the borough will remain in a yellow precautionary zone.
New Focus Zones
Manhattan - New Yellow Precautionary Zone - Click Here for Map
A new yellow zone will be established in the Washington Heights area, after meeting the metrics for yellow precautionary zone designation including on increasing 7-day positivity rates, number of cases, and hospitalizations.
Monroe County - Click Here for Map
The City of Rochester and surrounding areas will transition to an orange zone after meeting the positivity, new cases, and hospitalizations metrics for orange zone designation. The rest of the existing yellow zone will remain a Precautionary yellow zone.
Nassau County - Links to Maps Below
Yellow Precautionary zones will be established in and around Great Neck and Massapequa Park, after both areas have seen sustained positivity, new case rates, and hospitalizations that meet the metrics for yellow precautionary zone designation.
Onondaga County - Click Here for Map
The City of Syracuse and surrounding areas will transition to an orange precautionary zone after seeing continued increases in positivity and number of cases, as well as hospitalizations. The existing yellow zone will remain a yellow precautionary zone, and parts of it will be expanded.
Suffolk County - Links to Maps Below
Parts of Suffolk County including Hampton Bays and Riverhead will be designated as Precautionary Yellow Zones after meeting the positivity, new cases, and new hospitalization metrics for yellow zone designation.
The Governor noted that the positive testing rate in all focus areas under the state's Micro-Cluster strategy is 4.48 percent, and outside the focus zone areas is 2.73 percent. Within the focus areas, 39,209 test results were reported yesterday, yielding 1,755 positives. In the remainder of the state, not counting these focus areas, 152,280 test results were reported, yielding 4,151 positives. Full results for tests reported yesterday, the day prior, the current 7-day rolling average, and last two weeks is below:
STATEWIDE | 11/1- 11/7 % Positive | 11/8- 11/14 % Positive | 11/15- 11/21 % Positive | Current 7-day rolling average | Day Prior (11/21) % Positive | Yesterday (11/22) % Positive |
All focus area statewide % positive | 3.44% | 4.89% | 4.57% | 4.57% | 4.39% | 4.48% |
Statewide % positive with all focus areas included | 1.95% | 2.86% | 2.89% | 2.93% | 2.74% | 3.08% |
Statewide % positive without all focus areas included | 1.81% | 2.47% | 2.44% | 2.52% | 2.29% | 2.73% |
Micro-cluster zone 7-day average positivity rates for today, yesterday, the day before, last week, and the week prior is below:
FOCUS ZONE | 11/1- 11/7 % Positive | 11/8- 11/14 % Positive | 11/15- 11/21 % Positive | Day Prior 7-day Rolling Average | Yesterday 7-day Rolling Average | Current 7-day Rolling average |
Erie orange-zone focus area % positive | 4.53% | 7.22% | 7.30% | 7.25% | 7.30% | 7.20% |
Erie Yellow-zone focus area % positive | 4.64% | 5.34% | 7.36% | 7.58% | 7.36% | 7.53% |
Niagara Yellow -zone focus area % positive | 4.29% | 5.10% | 4.44% | 4.30% | 4.44% | 4.53% |
Monroe Yellow-zone focus area % positive | 4.06% | 5.54% | 3.80% | 3.96% | 3.80% | 3.76% |
Onondaga Yellow-zone focus area % positive | 4.68% | 6.58% | 5.12% | 5.65% | 5.12% | 4.99% |
Queens Kew Garden Hills/Forest Hills/Astoria yellow-zone focus area % positive | 2.11% | 3.40% | 3.40% | 3.32% | 3.40% | 3.31% |
Bronx East Yellow-zone focus area % positive | 2.70% | 3.81% | 3.52% | 3.49% | 3.52% | 3.73% |
Bronx West Yellow -zone focus area % positive | 2.79% | 3.80% | 4.70% | 4.72% | 4.70% | 4.73% |
Brooklyn Yellow-zone focus area % positive | 3.33% | 3.92% | 3.70% | 3.72% | 3.70% | 3.85% |
Rockland Yellow-zone focus area % positive | 3.46% | 3.55% | 3.39% | 3.30% | 3.39% | 3.29% |
Chemung Orange-zone focus area % positive | 6.45% | 4.59% | 4.71% | 4.15% | 4.71% | 5.20% |
Staten Island Yellow-zone focus area % positive | 3.00% | 4.33% | 4.15% | 3.99% | 4.15% | 4.28% |
Tioga Yellow-zone focus area % positive | 10.03% | 10.81% | 5.60% | 5.90% | 5.60% | 5.47% |
Orange Middletown - Yellow-zone focus area % positive | 2.95% | 3.81% | 5.41% | 4.94% | 5.41% | 3.84% |
Orange Newburgh - Yellow-zone focus area % positive | 5.96% | 8.07% | 7.89% | 7.93% | 7.89% | 8.68% |
Westchester Peekskill - Yellow-zone focus area % positive | 5.16% | 10.36% | 7.15% | 7.70% | 7.15% | 7.29% |
Westchester Ossining - Yellow-zone focus area % positive | 5.00% | 9.88% | 10.22% | 10.37% | 10.22% | 10.11% |
Westchester Tarrytown - Yellow-zone focus area % positive | 4.26% | 8.47% | 8.27% | 8.56% | 8.27% | 8.80% |
Westchester Yonkers - Yellow-zone focus area % positive | 3.42% | 4.48% | 4.11% | 4.07% | 4.11% | 4.38% |
Westchester New Rochelle - Yellow-zone focus area % positive | 4.40% | 6.46% | 5.68% | 6.58% | 5.68% | 5.20% |
Westchester Port Chester Orange-zone focus area % positive | 7.46% | 9.34% | 7.59% | 7.09% | 7.59% | 7.94% |
- Patient Hospitalization - 2,724 (+162)
- Patients Newly Admitted - 363
- Hospital Counties - 52
- Number ICU - 545 (+43)
- Number ICU with Intubation - 249 (+15)
- Total Discharges - 83,501 (+194)
- Deaths - 33
- Total Deaths - 26,390
Each region's percentage of positive test results reported over the last three days is as follows:
REGION Totalfinder advanced finder plugin 1 12 3. | FRIDAY | SATURDAY | SUNDAY | CURRENT 7-DAY AVERAGE |
Capital Region | 2.4% | 2.0% | 2.0% | 2.28% |
Central New York | 3.5% | 2.6% | 3.6% | 3.43% |
Finger Lakes | 3.8% | 2.5% | 3.6% | 3.30% |
Long Island | 2.9% | 3.5% | 3.5% | 3.24% |
Mid-Hudson | 3.7% | 3.3% | 4.2% | 3.91% |
Mohawk Valley | 3.3% | 2.1% | 2.5% | 2.58% |
New York City | 2.6% | 2.3% | 2.5% | 2.57% |
North Country | 2.2% | 1.3% | 1.7% | 1.73% |
Southern Tier | 1.2% | 1.5% | 1.8% | 1.23% |
Western New York | 5.7% | 6.0% | 5.5% | 5.11% |
Each New York City borough's percentage of positive test results reported over the last three days is as follows:
BOROUGH | FRIDAY | SATURDAY | SUNDAY | CURRENT 7-DAY AVERAGE |
Bronx | 3.9% | 3.3% | 3.4% | 3.39% |
Brooklyn | 2.4% | 2.0% | 2.7% | 2.33% |
Manhattan | 2.0% | 1.4% | 1.4% | 1.78% |
Queens | 2.4% | 2.9% | 2.9% | 2.81% |
Staten Island | 3.8% | 4.4% | 4.2% | 4.15% |
Of the 602,120 total individuals who tested positive for the virus, the geographic breakdown is as follows:
County | Total Positive | New Positive |
Albany | 5,148 | 40 |
Allegany | 856 | 10 |
Broome | 4,927 | 68 |
Cattaraugus | 901 | 24 |
Cayuga | 735 | 31 |
Chautauqua | 1,403 | 11 |
Chemung | 2,702 | 55 |
Chenango | 578 | 15 |
Clinton | 428 | 10 |
Columbia | 965 | 3 |
Cortland | 907 | 23 |
Delaware | 314 | 5 |
Dutchess | 6,728 | 94 |
Erie | 21,572 | 497 |
Essex | 250 | 2 |
Franklin | 239 | 8 |
Fulton | 454 | 22 |
Genesee | 800 | 42 |
Greene | 633 | 7 |
Hamilton | 39 | 2 |
Herkimer | 582 | 7 |
Jefferson | 435 | 17 |
Lewis | 306 | 6 |
Livingston | 617 | 34 |
Madison | 821 | 24 |
Monroe | 12,771 | 365 |
Montgomery | 396 | 18 |
Nassau | 57,127 | 421 |
Niagara | 3,213 | 61 |
NYC | 296,547 | 1,782 |
Oneida | 4,210 | 129 |
Onondaga | 9,362 | 202 |
Ontario | 1,129 | 42 |
Orange | 15,987 | 115 |
Orleans | 576 | 15 |
Oswego | 1,273 | 86 |
Otsego | 529 | 21 |
Putnam | 2,472 | 110 |
Rensselaer | 1,528 | 13 |
Rockland | 20,708 | 106 |
Saratoga | 1,969 | 65 |
Schenectady | 2,136 | 63 |
Schoharie | 176 | 7 |
Schuyler | 271 | 9 |
Seneca | 253 | 10 |
St. Lawrence | 678 | 11 |
Steuben | 1,599 | 42 |
Suffolk | 56,865 | 526 |
Sullivan | 2,093 | 15 |
Tioga | 1,043 | 19 |
Tompkins | 936 | 32 |
Ulster | 3,073 | 50 |
Warren | 543 | 1 |
Washington | 430 | 8 |
Wayne | 967 | 55 |
Westchester | 47,262 | 421 |
Wyoming | 419 | 18 |
Yates | 239 | 11 |
Yesterday, 33 New Yorkers died due to COVID-19 in New York State, bringing the total to 26,390. A geographic breakdown is as follows, by county of residence:
Deaths by County of Residence | |
County | New Deaths |
Allegany | 1 |
Bronx | 3 |
Chemung | 1 |
Erie | 10 |
Genesee | 1 |
Kings | 2 |
Lewis | 1 |
Madison | 1 |
Monroe | 3 |
Nassau | 1 |
Richmond | 3 |
Rockland | 1 |
Steuben | 1 |
Suffolk | 3 |
Tioga | 1 |
Contact the Governor's Press Office
Contact us by phone:
New York City: (212) 681 - 4640
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