Question

1. drag and drop the correct words to complete the sentences.

a ______ is the locus of points in a plane that are equidistant from a line and a point not on the line.
the ______ of a parabola is the point, along with a line not containing the point, which is used to generate a parabola.
the ______ of a parabola is the line, along with a point not on the line, which is used to generate a parabola.
the ______ of a parabola is the point of a parabola lying halfway between the directrix and focus.
choices
parabola
vertex
focus
name
directrix

1. use the equation $y = \frac{1}{2}(x - 3)^2 + 4$ to fill in the blanks.

opening ______
directrix $y =$ ______
focus ______
vertex ______

Explanation:

Part 1: Drag and Drop (Question 1)

• A parabola is defined as the set of points equidistant from a line (directrix) and a point (focus) not on the line.
• The focus is the point used (along with the directrix) to generate the parabola.
• The directrix is the line used (along with the focus) to generate the parabola.
• The vertex is the midpoint between the directrix and the focus on the parabola.

Answer:

s:

1. A \(\boldsymbol{\text{parabola}}\) is the locus of points in a plane that are equidistant from a line and a point not on the line.
2. The \(\boldsymbol{\text{focus}}\) of a parabola is the point, along with a line not containing the point, which is used to generate a parabola.
3. The \(\boldsymbol{\text{directrix}}\) of a parabola is the line, along with a point not on the line, which is used to generate a parabola.
4. The \(\boldsymbol{\text{vertex}}\) of a parabola is the point of a parabola lying halfway between the directrix and focus.

Part 2: Analyze the Parabola Equation \( y = \frac{1}{2}(x - 3)^2 + 4 \)

The standard form of a parabola that opens up/down is \( y = a(x - h)^2 + k \), where:

• Vertex: \( (h, k) \)
• Focus: \( (h, k + \frac{1}{4a}) \)
• Directrix: \( y = k - \frac{1}{4a} \)
• Direction: If \( a > 0 \), opens up; if \( a < 0 \), opens down.

Step 1: Determine the Vertex

For \( y = \frac{1}{2}(x - 3)^2 + 4 \), compare with \( y = a(x - h)^2 + k \):
\( h = 3 \), \( k = 4 \), \( a = \frac{1}{2} \).
Thus, the vertex is \( (3, 4) \).

Step 2: Determine the Direction of Opening

Since \( a = \frac{1}{2} > 0 \), the parabola opens upward.

Step 3: Calculate the Focus

Using the formula for the focus \( (h, k + \frac{1}{4a}) \):
Substitute \( h = 3 \), \( k = 4 \), \( a = \frac{1}{2} \):
\( \frac{1}{4a} = \frac{1}{4 \cdot \frac{1}{2}} = \frac{1}{2} \).
Thus, the focus is \( (3, 4 + \frac{1}{2}) = (3, \frac{9}{2}) \) (or \( (3, 4.5) \)).

Step 4: Calculate the Directrix

Using the formula for the directrix \( y = k - \frac{1}{4a} \):
Substitute \( k = 4 \), \( \frac{1}{4a} = \frac{1}{2} \):
\( y = 4 - \frac{1}{2} = \frac{7}{2} \) (or \( y = 3.5 \)).

Final Answers for Part 2:

• Opening: \(\boldsymbol{\text{upward}}\)
• Directrix: \( y = \boldsymbol{\frac{7}{2}} \) (or \( y = 3.5 \))
• Focus: \( \boldsymbol{(3, \frac{9}{2})} \) (or \( (3, 4.5) \))
• Vertex: \( \boldsymbol{(3, 4)} \)