Quiz worksheet on parabolas with questions about equations, vertex, focus, and directrix.

Let's solve each problem step by step.

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Problem 1:

Which correctly describes the parabola: \( y = -\frac{1}{2}(x + 3)^2 + 4 \)

#### Solution:
The given equation is in the vertex form of a parabola:
\[ y = a(x - h)^2 + k \]
where:
- \( (h, k) \) is the vertex,
- \( a \) determines the direction and width of the parabola.

For the given equation:
\[ y = -\frac{1}{2}(x + 3)^2 + 4 \]
we can identify:
- \( a = -\frac{1}{2} \),
- \( h = -3 \),
- \( k = 4 \).

The vertex is \( (-3, 4) \).

The sign of \( a \) determines the direction of the parabola:
- If \( a > 0 \), the parabola opens up.
- If \( a < 0 \), the parabola opens down.

Here, \( a = -\frac{1}{2} \), which is negative. Therefore, the parabola opens down.

#### Answer:
\[ \boxed{\text{C}} \]

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Problem 2:

Which correctly describes the parabola: \( y = 2(x - 1)^2 - 3 \)

#### Solution:
The given equation is also in the vertex form:
\[ y = a(x - h)^2 + k \]
where:
- \( (h, k) \) is the vertex,
- \( a \) determines the direction and width of the parabola.

For the given equation:
\[ y = 2(x - 1)^2 - 3 \]
we can identify:
- \( a = 2 \),
- \( h = 1 \),
- \( k = -3 \).

The vertex is \( (1, -3) \).

The sign of \( a \) determines the direction of the parabola:
- If \( a > 0 \), the parabola opens up.
- If \( a < 0 \), the parabola opens down.

Here, \( a = 2 \), which is positive. Therefore, the parabola opens up.

#### Answer:
\[ \boxed{\text{C}} \]

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Problem 3:

Which correctly describes the parabola: \( x = 7(y - 6)^2 - 2 \)

#### Solution:
The given equation is in the form:
\[ x = a(y - k)^2 + h \]
where:
- \( (h, k) \) is the vertex,
- \( a \) determines the direction and width of the parabola.

For the given equation:
\[ x = 7(y - 6)^2 - 2 \]
we can identify:
- \( a = 7 \),
- \( h = -2 \),
- \( k = 6 \).

The vertex is \( (-2, 6) \).

The sign of \( a \) determines the direction of the parabola:
- If \( a > 0 \), the parabola opens right.
- If \( a < 0 \), the parabola opens left.

Here, \( a = 7 \), which is positive. Therefore, the parabola opens right.

#### Answer:
\[ \boxed{\text{C}} \]

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Problem 4:

What is the directrix of this parabola?

#### Solution:
The given parabola is shown in the graph, and it opens downward. The focus is at \( (0, -3) \).

For a parabola that opens upward or downward, the standard form is:
\[ (x - h)^2 = 4p(y - k) \]
where:
- \( (h, k) \) is the vertex,
- \( p \) is the distance from the vertex to the focus (and also from the vertex to the directrix).

From the graph:
- The vertex is at \( (0, 0) \),
- The focus is at \( (0, -3) \).

The distance from the vertex to the focus is:
\[ p = -3 \]
(since the focus is below the vertex, \( p \) is negative).

The directrix is a horizontal line located at a distance \( |p| \) above the vertex. Since \( p = -3 \), the directrix is:
\[ y = 0 - (-3) = 3 \]

#### Answer:
\[ \boxed{\text{A}} \]

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Problem 5:

The focus is always inside the parabola.

#### Solution:
This statement is true for all parabolas. By definition, the focus of a parabola is a point inside the parabola, and it is used to define the parabola along with the directrix.

#### Answer:
\[ \boxed{\text{B}} \]

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Problem 6:

Identify the vertex.
\[ y = -(x + 9)^2 - 3 \]

#### Solution:
The given equation is in the vertex form:
\[ y = a(x - h)^2 + k \]
where:
- \( (h, k) \) is the vertex,
- \( a \) determines the direction and width of the parabola.

For the given equation:
\[ y = -(x + 9)^2 - 3 \]
we can rewrite it as:
\[ y = -1(x - (-9))^2 + (-3) \]

From this, we identify:
- \( h = -9 \),
- \( k = -3 \).

The vertex is \( (-9, -3) \).

#### Answer:
\[ \boxed{\text{A}} \]

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Final Answers:

1. \( \boxed{\text{C}} \)
2. \( \boxed{\text{C}} \)
3. \( \boxed{\text{C}} \)
4. \( \boxed{\text{A}} \)
5. \( \boxed{\text{B}} \)
6. \( \boxed{\text{A}} \)