91Ó°ÊÓ

A parabola is a U-shaped curve that represents the graph of a quadratic equation. In standard form, a quadratic equation looks like this: \( y = ax^2 + bx + c \). The key feature of a parabola is that it can open either upwards or downwards, depending on the sign of the coefficient \( a \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards, creating an upside-down U shape.

For our equation, \( y = -x^2 + 2x + 3 \), the coefficient of \( x^2 \) is negative, hence the parabola opens downwards. In addition to opening direction, parabolas have some unique points and lines that help in understanding their shape and behavior. These include intercepts and the vertex, both of which we will explore in detail.

Intercepts are the points where the parabola crosses the x-axis and y-axis.

• Y-intercept: This is where the graph crosses the y-axis. To find this point, set \( x = 0 \) in the equation and solve for \( y \). In our case: \( y = -0^2 + 2(0) + 3 = 3 \), so the y-intercept is (0, 3).
• X-intercepts: These are the points where the graph crosses the x-axis. To find these points, set \( y = 0 \) and solve for \( x \). For our equation: \( 0 = -x^2 + 2x + 3 \). Rearrange and solve the quadratic equation: \( x^2 - 2x - 3 = 0 \). Using the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \), we get two solutions, \( x = 3 \) and \( x = -1 \), so the x-intercepts are at (3, 0) and (-1, 0).

Understanding the intercepts is crucial because they provide key points through which the graph of the quadratic equation passes. This helps in accurately sketching the parabola.

The vertex of a parabola is its highest or lowest point, depending on the direction it opens. For downward-opening parabolas, like ours, the vertex is the maximum point. The vertex can be found using the formula for the x-coordinate: \( x = -\frac{b}{2a} \).Using this formula for our equation \( y = -x^2 + 2x + 3 \), we get: \( x = -\frac{2}{2(-1)} = 1 \).Substitute \( x = 1 \) back into the original equation to get the y-coordinate of the vertex: \( y = -1^2 + 2(1) + 3 = 4 \).Therefore, the vertex of our parabola is at the point (1, 4).

• The vertex form of a quadratic equation can be written as: \( y = a(x-h)^2 + k \), where (h, k) is the vertex.

Knowing the vertex is important for understanding the shape and position of the parabola in the coordinate plane. It also helps in sketching the graph accurately.