91Ó°ÊÓ

To find the vertex of a parabola in the standard form equation: \( y = ax^2 + bx + c \), we use a specific formula. The vertex is a crucial point where the parabola changes direction.

The x-coordinate of the vertex is given by:
\( x = -\frac{b}{2a} \).

For the equation \( y = -x^2 + 4x + 3 \):

• \( a = -1 \)
• \( b = 4 \)

Plugging these values into the formula:

\( x = -\frac{4}{2(-1)} = 2 \).

Next, we find the y-coordinate by substituting \( x = 2 \) back into the original equation:
\( y = - (2)^2 + 4(2) + 3 = -4 + 8 + 3 = 7 \).

Therefore, the vertex of the parabola is at (2, 7). This point is the highest or lowest point on the parabola, depending on its direction.

The focus of a parabola is a fixed point used to define the parabola. In a downward-opening parabola like\( y = -x^2 + 4x + 3 \), the focus lies below the vertex. To find the focus, we need to determine the distance \( p \) from the vertex to the focus.

We convert our quadratic equation to the form \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex. For our parabola:

• \( h = 2 \)
• \( k = 7 \)

The equation can be rewritten as:
\( y = -(x - 2)^2 + 7 \).

By comparing with \( y = k - \frac{1}{4p}(x-h)^2 \), we find:

\( -\frac{1}{4p} = -1 \).

Solve for \( p \):
\( \frac{1}{4p} = 1 \) implies \( 4p = 1 \) or \( p = \frac{1}{4} \).

Therefore, the focus lies \( \frac{1}{4} \) units below the vertex:
Focus = (2, 6.75). The focus helps in plotting the accuracy of the parabola and determining its geometric properties.

The directrix of a parabola is a line that is perpendicular to the axis of symmetry of the parabola. It's used, alongside the focus, to define the set of points that form the parabola. For a parabola that opens downward like \( y = -x^2 + 4x + 3 \), the directrix is above the vertex.

We already determined the distance \( p \) is \( \frac{1}{4} \) units. This is used to find the equation of the directrix.

• Since the vertex is at (2, 7)
• And \( p = \frac{1}{4} \)

To find the directrix, move \( \frac{1}{4} \) units above the vertex:
Directrix = \( y = 7 + \frac{1}{4} = 7.25 \).

This line helps in understanding how the parabola is defined and its orientation in the Cartesian plane.

To know the direction a parabola opens, we look at the coefficient \( a \) in the standard form equation \( y = ax^2 + bx + c \).

- If \( a > 0 \), the parabola opens upward.
- If \( a < 0 \), the parabola opens downward.

For the equation \( y = -x^2 + 4x + 3 \), the coefficient of \( x^2 \) (i.e., \( a \)) is -1:

Since \( a < 0 \)
This tells us the parabola opens downward.

The direction is essential as it tells us if the vertex is a maximum or a minimum point on the graph. In this case, the vertex (2, 7) is the maximum point, meaning the parabola extends downward from the vertex.