Graphing

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The length of line segment $\overline{AB}$ is 12 units. If point A is located at , what is a possible location for point B?

$(14, 5+2\sqrt{11})$

$(2\sqrt{11}, 10)$

$(4+2\sqrt{22}, 15)$

Explanation

To answer this question, we will have to manipulate the distance formula:

$d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}$

To get rid of the square root, we can square both sides:

$d^{2}=(x_{1}-x_{2})^{2}}+(y_{1}-y_{2})^{2}}$

and plug in the information given in the question.

$12^{2}=(4-x_{2})^{2}}+(5-y_{2})^{2}}$

$144=(4-x_{2})^{2}}+(5-y_{2})^{2}}$

At this point we can simply plug in the possible values to determine which combination of coordinates will make the equation above true.

$144=(4-14)^{2}}+(5-(5+2{\sqrt{11}}))^{2}}$ $=(-10)^{2}+(2\sqrt{11})^{2}=100+44=144$

Thus the correct coordinate is,

$(14, 5+2\sqrt{11})$ .

The points and are plotted on a quadrant. Which graph depicts the correct points?

Explanation

A quadrant is set up in such a way that the positive numbers are to the right and top, while the negative numbers are to the left and bottom. The x-value (the first number in the ordered pair) is the distance left or right from the center. The y-value (the second number in the ordered pair) is the distance above or below the axis.

will be units to the right, and units up.

will be units to the left, and units up.

The length of line segment $\overline{AB}$ is 12 units. If point A is located at , what is a possible location for point B?

$(14, 5+2\sqrt{11})$

$(2\sqrt{11}, 10)$

$(4+2\sqrt{22}, 15)$

Explanation

To answer this question, we will have to manipulate the distance formula:

$d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}$

To get rid of the square root, we can square both sides:

$d^{2}=(x_{1}-x_{2})^{2}}+(y_{1}-y_{2})^{2}}$

and plug in the information given in the question.

$12^{2}=(4-x_{2})^{2}}+(5-y_{2})^{2}}$

$144=(4-x_{2})^{2}}+(5-y_{2})^{2}}$

At this point we can simply plug in the possible values to determine which combination of coordinates will make the equation above true.

$144=(4-14)^{2}}+(5-(5+2{\sqrt{11}}))^{2}}$ $=(-10)^{2}+(2\sqrt{11})^{2}=100+44=144$

Thus the correct coordinate is,

$(14, 5+2\sqrt{11})$ .

Find the point that corresponds to the following ordered pair:

Explanation

In order to get to the point , start at the origin and move right unit and up units.

This is point .

Find the point that corresponds to the following ordered pair:

Explanation

In order to get to the point , start at the origin and move right unit and up units.

This is point .

The points and are plotted on a quadrant. Which graph depicts the correct points?

Explanation

will be units to the right, and units up.

will be units to the left, and units up.

Find the point that corresponds to the following ordered pair:

Explanation

In order to get to the point , start at the origin and move right units and down units.

This is point .

For the graph below, match the graph b with one of the following equations:

$y= x^{2}$

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

$y = (x+3)^{2}$

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

Parabola

$y = (x+3)^{2}$

$y = x^{2}$

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

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

None of the above

Explanation

Starting with $y = x^{2}$

$(x-2)^{2}$ moves the parabola $x^{2}$ by units to the right.

Similarly $(x+3)^{2}$ moves the parabola by units to the left.

Hence the correct answer is option .

What is the equation of a parabola with vertex and -intercept ?

$y = -\frac{1}{2} x^{2} +4x - 7$

$y = -\frac{1}{2} x^{2} -4x - 7$

$y = \frac{1}{2} x^{2} -4x - 7$

$y = \frac{1}{2} x^{2} -8x - 7$

$y = \frac{1}{2} x^{2} +8x - 7$

Explanation

From the vertex, we know that the equation of the parabola will take the form $y = a (x - 4) ^{2}+ 1$ for some .

To calculate that , we plug in the values from the other point we are given, , and solve for :

$y = a (x - 4)^{2} + 1$

$-7= a \cdot (0 - 4)^{2} + 1$

$-7= a \cdot (- 4)^{2} + 1$

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

Now the equation is $y = -\frac{1}{2} (x - 4)^{2} + 1$ . This is not an answer choice, so we need to rewrite it in some way.

Expand the squared term:

$y = -\frac{1}{2} (x^{2} -8x + 16) + 1$

Distribute the fraction through the parentheses:

$y = -\frac{1}{2} x^{2} +4x - 8 + 1$

Combine like terms:

$y = -\frac{1}{2} x^{2} +4x - 7$

Find the point that corresponds to the following ordered pair:

Explanation

In order to get to the point , start at the origin and move right units and down units.

This is point .

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