Graph g(x)=x^2-2x-3

Graph g(x)=x^2-2x-3
Find the properties of the given parabola.
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Rewrite the equation in vertex form.
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Complete the square for .
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Use the form , to find the values of , , and .
Consider the vertex form of a parabola.
Substitute the values of and into the formula .
Simplify the right side.
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Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Multiply by .
Find the value of using the formula .
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Simplify each term.
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Raise to the power of .
Multiply by .
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Multiply by .
Subtract from .
Substitute the values of , , and into the vertex form .
Set equal to the new right side.
Use the vertex form, , to determine the values of , , and .
Since the value of is positive, the parabola opens up.
Opens Up
Find the vertex .
Find , the distance from the vertex to the focus.
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Find the distance from the vertex to a focus of the parabola by using the following formula.
Substitute the value of into the formula.
Cancel the common factor of .
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Cancel the common factor.
Rewrite the expression.
Find the focus.
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The focus of a parabola can be found by adding to the y-coordinate if the parabola opens up or down.
Substitute the known values of , , and into the formula and simplify.
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
Find the directrix.
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The directrix of a parabola is the horizontal line found by subtracting from the y-coordinate of the vertex if the parabola opens up or down.
Substitute the known values of and into the formula and simplify.
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
Select a few values, and plug them into the equation to find the corresponding values. The values should be selected around the vertex.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raising to any positive power yields .
Multiply by .
Simplify by adding and subtracting.
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Add and .
Subtract from .
The final answer is .
The value at is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of .
Multiply by .
Simplify by adding and subtracting.
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Add and .
Subtract from .
The final answer is .
The value at is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
Subtract from .
The final answer is .
The value at is .
Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
Subtract from .
The final answer is .
The value at is .
Graph the parabola using its properties and the selected points.
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex:
Focus:
Axis of Symmetry:
Directrix:
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Name

Name two billion seventy-five million one hundred ninety-seven thousand eighty-two

Interesting facts

  • 2075197082 has 16 divisors, whose sum is 3559238592
  • The reverse of 2075197082 is 2807915702
  • Previous prime number is 2083

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 41
  • Digital Root 5

Name

Name six hundred seven million eight hundred forty thousand ninety-three

Interesting facts

  • 607840093 has 16 divisors, whose sum is 713007360
  • The reverse of 607840093 is 390048706
  • Previous prime number is 137

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 37
  • Digital Root 1

Name

Name one hundred ten million five hundred thirty-eight thousand six hundred forty-five

Interesting facts

  • 110538645 has 16 divisors, whose sum is 139475952
  • The reverse of 110538645 is 546835011
  • Previous prime number is 293

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 33
  • Digital Root 6