A dilation is the stretching or shrinking of a figure.

A rotation is the turning of a a figure about a point.

A reflection is the flipping of a figure across a line.

A translation is is the sliding of a figure in a direction.

With a translation, the image is not only congruent to its original size and shape, but its orientation remains the same. A translation fits this figure best because the shape seems to move upward and rightward without changing size, shape, or orientation.

The median of a set of eleven data values - an odd number - is the value that appears in the middle when the values are ranked. For 5 to be the median, 5 must be in the middle - that is, five values must appear before 5, and five values must appear after 8.

We can answer this question by looking at three cases.

Case 1:

Without loss of generality, assume ; this reasoning holds for any lesser value of . The data set becomes

,

and the median is 4.

Case 2:

The data set becomes

The middle value - the median - is 5.

Case 3:

Without loss of generality, assume ; this reasoning holds for any greater value of . The data set becomes

Again, the median is 5.

Therefore, we can set equal to 5, 6, 7, 8, 9, or 10 - any of six different values - and make the median of the set 5.

An expression with a radical expression in the denominator is not simplified, so to simplify, it is necessary to rationalize the denominator. This is accomplished by multiplying both numerator and denominator by the given square root, , as follows:

The expression can be simplified further by dividing the numbers outside the radical by greatest common factor 5:

This is the correct response.

To simplify a radical expression, first find the prime factorization of the radicand, which is 32 here.

Pair up like factors, then apply the Product of Radicals Property:

,

the simplest form of the radical.

To simplify a radical expression, first find the prime factorization of the radicand, which is 40 here.

Pair up like factors, then apply the Product of Radicals Property:

,

the simplest form of the radical.

To simplify a radical expression, first find the prime factorization of the radicand, which is 120 here.

Pair up like factors, then apply the Product of Radicals Property:

,

the simplest form of the radical.

Examine the figure below:

If we connect the horizontal line with the line along the rotated nine at right, we see that it is the result of a one-eighth turn clockwise; the angle between them .

The question asks, "Which of the following functions best models the number of paintings she will have after days?"

From this, you know that the variable represents the number of days, and that represents the number of paintings she makes as a function of days spent working.

If it takes 3 days to make a painting, each day results in paintings. Therefore, we have a linear relationship with slope .

Additionally, she begins with 6 paintings. Therefore, even when zero days are spent working on paintings, she will have 6 paintings. In other words, . This means the y-intercept is 6.

As a result, the function will be

which can be rewritten as

Use the zeroes of the graph to determine the matching function. Zeroes are values of x where . In other words, they are points on the graph where the curve touches zero.

Visually, you can see that the curve crosses the x-axis when , , and . Therefore, you need to look for a function that will equal zero at these x values.

A function with a factor of will equal zero when , because the factor of will equal zero. The matching factors for the other two zeroes, and , are and , respectively.

The answer choice has all of these factors, but it is not the answer because it has an additional zero that would be visible on the graph. Notice it has a factor of , which results in a zero at . This additional zero that isn't present in the graph indicates that this cannot be matching function.

is the answer because it has all of the required factors and, as a result, the required zeroes, while not having additional zeroes. Notice that the constant coefficient of negative 2 does not affect where the zeroes are.

To perform a dilation with center and scale factor , find the midpoints of the segments connecting to each point, and connect those points. We can simplify the process by finding the midpoints of , , and , and naming them , , and , respectively; , the image of center , is just itself. The figure is below:

Now, do the same thing to the new square, but with as the center. The figure is below:

The final image, relative to the original square, is below: