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Understanding the vertex form of a quadratic function is essential for easily determining crucial properties of a parabola, such as the vertex, direction, and intercepts. The vertex form of a quadratic equation is given as \( f(x) = a(x-h)^2 + k \). In this expression:

• \( a \) indicates the degree of vertical stretch or compression and the direction of the parabola.
• The \( (h, k) \) provides the coordinates for the vertex of the parabola.

For the function \( f(x) = -2(x-3)^2 + 18 \), the vertex form directly gives us the vertex at (3, 18). In practice, converting a function from standard form \( ax^2 + bx + c \) to vertex form often involves a process called completing the square. This allows us to easily locate the vertex by identifying \( h \) and \( k \) in the expression.

The direction in which a parabola opens depends on the coefficient \( a \) in the quadratic function. When the quadratic equation is in the form \( f(x) = ax^2 + bx + c \), the sign of \( a \) determines the parabola's direction:

• If \( a > 0 \), the parabola opens upward, resembling a U-shape.
• If \( a < 0 \), the parabola opens downward, similar to an upside-down U.

In our example function \( f(x) = -2x^2 + 12x \), the coefficient \( a = -2 \), which means the parabola opens downwards. This tells us that the vertex is a maximum point on the graph. The visual representation of the parabola will show closing arms directing downwards, indicating decreasing values of \( f(x) \) as we move away from the vertex.

The x-intercepts are the points where the graph of the function crosses the x-axis. These points occur when \( f(x) = 0 \). To find the x-intercepts, solve the equation for \( x \).In our quadratic function, first, we express it as the vertex form, \( f(x) = -2(x - 3)^2 + 18 \), and set it to zero to find the intercepts:\[ 0 = -2(x-3)^2 + 18 \]Solving this, we find two values for \( x \):

• \( x = 0 \)
• \( x = 6 \)

Thus, our x-intercepts are at points (0,0) and (6,0). These intercepts indicate where the graph of the function touches or crosses the x-axis, providing two points through which the parabola passes.

The y-intercept of a quadratic function is where the graph intersects the y-axis, which occurs when \( x = 0 \). This point represents the value of the function when no horizontal distance from the origin is considered.To find the y-intercept for the function \( f(x) = -2x^2 + 12x \), substitute \( x = 0 \):\[ f(0) = -2(0)^2 + 12(0) \]This calculation simplifies to 0, so the y-intercept is at (0, 0). Interestingly, for this particular function, the y-intercept coincides with one of the x-intercepts. Intercepts are valuable because they provide real-number solutions that help plot significant points on the graph, aiding in the construction and understanding of the parabola's shape.