A polynomial function with zeroes 3 and 7 has as its factors and . The function is given to be quadratic, so this function is

.

Apply the FOIL method to rewrite the polynomial:

Collect like terms:

,

the correct choice.

To solve for in a literal equation, use the properties of algebra to isolate on one side, just as if you were solving a regular equation.

First, take the reciprocal of both sides:

Multiply both sides by :

Distribute on the right:

Subtract 1 from both sides, rewriting 1 as to facilitate subtraction:

,

the correct response.

The first step to solving an equation where is in a radical is to isolate the radical. To do this, we need to subtract the 5 from both sides.

Now that the radical is isolated, clear the radical by raising both sides to the power of 3. Note:

Now we want to isolate the term. First, subtract the 5 from both sides.

Finally, divide both sides by to solve for .

Given a quadratic function

the vertex will always be

.

Thus, since our function is

, , and .

We plug these variables into the formula to get the vertex as

.

Hence, the vertex of

is

.

Solve for in the equation

by isolating on the left side. Do this by reversing the operations in the reverse of the order of operations.

First, subtract 17 from both sides:

Now, divide both sides by 2:

One way to find 25% of this value is to multiply 41 by 25 and divide by 100:

,

the correct choice.

The range of a function is the set of all possible values of over its domain.

Since this function is piecewise-defined, it is necessary to examine both parts of the function and extract the range of each.

For , it holds that , so

,

or

.

For , it holds that , so

,

or

The overall range of is the union of these sets, or .

Step 1: Rearrange the terms from highest power to lowest power.

We will get: .

Step 2: We count the second term from starting from the left since it is the second highest term in the rearranged expression.

Step 3: Isolate the term.

The second term is

Step 4: Find the coefficient. The coefficient of a term is considered as the number before any variables. In this case, the coefficient is .

So, the answer is .

Solve for in the equation

by isolating on the left side. Do this by reversing the operations in the reverse of the order of operations.

First, subtract 17 from both sides:

Now, divide both sides by 2:

One way to find 25% of this value is to multiply 41 by 25 and divide by 100:

,

the correct choice.

Given a quadratic function

the vertex will always be

.

Thus, since our function is

, , and .

We plug these variables into the formula to get the vertex as

.

Hence, the vertex of

is

.

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