Graphing Parabolas (Focus and Directrix) 1

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Question 1 of 5

1. Question
Which of the following will most likely be the graph of

$x^{2} = 4 a y$ $x^2=4ay$
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In a quadratic equation, if the coefficient of $y$ $y$ is positive, the parabola will be concave up. If it is negative, the parabola will be concave down

Notice that there is no constant term (term without a variable).

$x^{2}$ $x^2$ $=$ $=$ $4 a y$ $4ay$

This means that the vertex is at $(0, 0)$ $(0,0)$ , the origin

Also, take note of the following components:

Focus $(0, a)$ $(0,a)$

this point lies in the parabola’s axis of symmetry and is at equal distance with all points on the parabola

Directrix $y = - a$ $y=-a$

this line lies in the opposite direction of the parabola and is at equal distance with all points on the parabola

Next, check the sign of the coefficient of $y$ $y$ .

$x^{2}$ $x^2$ $=$ $=$ $4 a$ $4a$ $y$ $y$

Hence, the parabola is concave up.

Therefore, it would make sense to say that the graph for $x^{2} = 4 a y$ $x^2=4ay$ is
Question 2 of 5

2. Question
Sketch the graph of $x^{2} = 16 y$ $x^2=16y$
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Standard Form (Concave Up)

$x^{2} = 4$ $x^2=4$ $a$ $a$ $y$ $y$

Focus

$(0,$ $(0,$ $a$ $a$ $)$ $)$

Standard Form (Concave Down)

$x^{2} = - 4$ $x^2=-4$ $a$ $a$ $y$ $y$

Focus

$(0, -$ $(0,-$ $a$ $a$ $)$ $)$

First, determine which standard form applies to the given equation.

$x^{2}$ $x^2$ $=$ $=$ $16 y$ $16y$

Since the coefficient of $y$ $y$ is positive, use $x^{2} = 4$ $x^2=4$ $a$ $a$ $y$ $y$

This also means that the parabola will be concave up

Solve for the focal length, $a$ $a$ , by comparing the given equation to the standard form

$x^{2}$ $x^2$ $=$ $=$ $4 a$ $4a$ $y$ $y$

$x^{2}$ $x^2$ $=$ $=$ $16$ $16$ $y$ $y$

$4 a$ $4a$ $=$ $=$ $16$ $16$

$4 a$ $4a$ $\div 4$ $divide4$ $=$ $=$ $16$ $16$ $\div 4$ $divide4$ Divide both sides by $4$ $4$

$a$ $a$ $=$ $=$ $4$ $4$

Identify the focus and directrix.

Focus:

$(0,$ $(0,$ $a$ $a$ $)$ $)$ becomes $(0,$ $(0,$ $4$ $4$ $)$ $)$

Directrix:

$y = -$ $y=-$ $a$ $a$ becomes $y = -$ $y=-$ $4$ $4$

Start graphing the parabola by plotting the focus and directrix

Finally, draw a parabola that is concave up
Question 3 of 5

3. Question
Sketch the graph of $x^{2} = - 8 y$ $x^2=-8y$
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Standard Form (Concave Up)

$x^{2} = 4$ $x^2=4$ $a$ $a$ $y$ $y$

Focus

$(0,$ $(0,$ $a$ $a$ $)$ $)$

Standard Form (Concave Down)

$x^{2} = - 4$ $x^2=-4$ $a$ $a$ $y$ $y$

Focus

$(0, -$ $(0,-$ $a$ $a$ $)$ $)$

First, determine which standard form applies to the given equation.

$x^{2}$ $x^2$ $=$ $=$ $- 8 y$ $-8y$

Since the coefficient of $y$ $y$ is negative, use $x^{2} = - 4$ $x^2=-4$ $a$ $a$ $y$ $y$

This also means that the parabola will be concave down

Solve for the focal length, $a$ $a$ , by comparing the given equation to the standard form

$x^{2}$ $x^2$ $=$ $=$ $- 4 a$ $-4a$ $y$ $y$

$x^{2}$ $x^2$ $=$ $=$ $- 8$ $-8$ $y$ $y$

$- 4 a$ $-4a$ $=$ $=$ $- 8$ $-8$

$- 4 a$ $-4a$ $\div - 4$ $divide-4$ $=$ $=$ $- 8$ $-8$ $\div - 4$ $divide-4$ Divide both sides by $- 4$ $-4$

$a$ $a$ $=$ $=$ $2$ $2$

Identify the focus and directrix.

Focus:

$(0, -$ $(0,-$ $a$ $a$ $)$ $)$ becomes $(0, -$ $(0,-$ $2$ $2$ $)$ $)$

Directrix:

$y =$ $y=$ $a$ $a$ becomes $y =$ $y=$ $2$ $2$

Start graphing the parabola by plotting the focus and directrix

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Finally, draw a parabola that is concave down
Question 4 of 5

4. Question
Sketch the graph of $y^{2} = 20 x$ $y^2=20x$
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Standard Form (Concave Right)

$y^{2} = 4$ $y^2=4$ $a$ $a$ $x$ $x$

Focus

$($ $($ $a$ $a$ $, 0)$ $,0)$

Standard Form (Concave Left)

$y^{2} = - 4$ $y^2=-4$ $a$ $a$ $x$ $x$

Focus

$(-$ $(-$ $a$ $a$ $, 0)$ $,0)$

First, determine which standard form applies to the given equation.

$y^{2}$ $y^2$ $=$ $=$ $20 x$ $20x$

Since the coefficient of $x$ $x$ is positive, use $y^{2} = 4$ $y^2=4$ $a$ $a$ $x$ $x$

This also means that the parabola will be concave right

Solve for the focal length, $a$ $a$ , by comparing the given equation to the standard form

$y^{2}$ $y^2$ $=$ $=$ $4 a$ $4a$ $x$ $x$

$y^{2}$ $y^2$ $=$ $=$ $20$ $20$ $x$ $x$

$4 a$ $4a$ $=$ $=$ $20$ $20$

$4 a$ $4a$ $\div 4$ $divide4$ $=$ $=$ $20$ $20$ $\div 4$ $divide4$ Divide both sides by $4$ $4$

$a$ $a$ $=$ $=$ $5$ $5$

Identify the focus and directrix.

Focus:

$($ $($ $a$ $a$ $, 0)$ $,0)$ becomes $($ $($ $5$ $5$ $, 0)$ $,0)$

Directrix:

$x = -$ $x=-$ $a$ $a$ becomes $x = -$ $x=-$ $5$ $5$

Start graphing the parabola by plotting the focus and directrix

Finally, draw a parabola that is concave right
Question 5 of 5

5. Question
Sketch the graph of $y^{2} = - 2 x$ $y^2=-2x$
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Standard Form (Concave Right)

$y^{2} = 4$ $y^2=4$ $a$ $a$ $x$ $x$

Focus

$($ $($ $a$ $a$ $, 0)$ $,0)$

Standard Form (Concave Left)

$y^{2} = - 4$ $y^2=-4$ $a$ $a$ $x$ $x$

Focus

$(-$ $(-$ $a$ $a$ $, 0)$ $,0)$

First, determine which standard form applies to the given equation.

$y^{2}$ $y^2$ $=$ $=$ $- 2 x$ $-2x$

Since the coefficient of $x$ $x$ is negative, use $y^{2} = - 4$ $y^2=-4$ $a$ $a$ $x$ $x$

This also means that the parabola will be concave left

Solve for the focal length, $a$ $a$ , by comparing the given equation to the standard form

$y^{2}$ $y^2$ $=$ $=$ $- 4 a$ $-4a$ $x$ $x$

$y^{2}$ $y^2$ $=$ $=$ $- 2$ $-2$ $x$ $x$

$- 4 a$ $-4a$ $=$ $=$ $- 2$ $-2$

$- 4 a$ $-4a$ $\div - 4$ $divide-4$ $=$ $=$ $- 2$ $-2$ $\div - 4$ $divide-4$ Divide both sides by $- 4$ $-4$

$a$ $a$ $=$ $=$ $\frac{1}{2}$ $1/2$

Identify the focus and directrix.

Focus:

$(-$ $(-$ $a$ $a$ $, 0)$ $,0)$ becomes $(-$ $(-$ $\frac{1}{2}$ $1/2$ $, 0)$ $,0)$

Directrix:

$x =$ $x=$ $a$ $a$ becomes $x =$ $x=$ $\frac{1}{2}$ $1/2$

Start graphing the parabola by plotting the focus and directrix

Finally, draw a parabola that is concave left

Graphing Parabolas (Focus and Directrix) 1

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Quiz summary

Information

1. Question

Hint

Well Done!

2. Question

Hint

Nice Job!

Standard Form (Concave Up)

Focus

Standard Form (Concave Down)

Focus

3. Question

Hint

Excellent!

Standard Form (Concave Up)

Focus

Standard Form (Concave Down)

Focus

4. Question

Hint

Fantastic!

Standard Form (Concave Right)

Focus

Standard Form (Concave Left)

Focus

5. Question

Hint

Keep Going!

Standard Form (Concave Right)

Focus

Standard Form (Concave Left)

Focus

Quizzes

$x^{2}$ $x^2$	$=$ $=$	$4 a$ $4a$ $y$ $y$
$x^{2}$ $x^2$	$=$ $=$	$16$ $16$ $y$ $y$

$4 a$ $4a$	$=$ $=$	$16$ $16$
$4 a$ $4a$ $\div 4$ $divide4$	$=$ $=$	$16$ $16$ $\div 4$ $divide4$	Divide both sides by $4$ $4$
$a$ $a$	$=$ $=$	$4$ $4$

$x^{2}$ $x^2$	$=$ $=$	$- 4 a$ $-4a$ $y$ $y$
$x^{2}$ $x^2$	$=$ $=$	$- 8$ $-8$ $y$ $y$

$- 4 a$ $-4a$	$=$ $=$	$- 8$ $-8$
$- 4 a$ $-4a$ $\div - 4$ $divide-4$	$=$ $=$	$- 8$ $-8$ $\div - 4$ $divide-4$	Divide both sides by $- 4$ $-4$
$a$ $a$	$=$ $=$	$2$ $2$

$y^{2}$ $y^2$	$=$ $=$	$4 a$ $4a$ $x$ $x$
$y^{2}$ $y^2$	$=$ $=$	$20$ $20$ $x$ $x$

$4 a$ $4a$	$=$ $=$	$20$ $20$
$4 a$ $4a$ $\div 4$ $divide4$	$=$ $=$	$20$ $20$ $\div 4$ $divide4$	Divide both sides by $4$ $4$
$a$ $a$	$=$ $=$	$5$ $5$

$y^{2}$ $y^2$	$=$ $=$	$- 4 a$ $-4a$ $x$ $x$
$y^{2}$ $y^2$	$=$ $=$	$- 2$ $-2$ $x$ $x$

$- 4 a$ $-4a$	$=$ $=$	$- 2$ $-2$
$- 4 a$ $-4a$ $\div - 4$ $divide-4$	$=$ $=$	$- 2$ $-2$ $\div - 4$ $divide-4$	Divide both sides by $- 4$ $-4$

$a$ $a$	$=$ $=$	$\frac{1}{2}$ $1/2$