91Ó°ÊÓ

In any quadratic function of the form \(f(x) = ax^2 + bx + c\), the vertex is a crucial point that represents either the highest or lowest point on the graph, depending on the parabola's orientation. To find the vertex, we use the formula for the x-coordinate: \(-\frac{b}{2a}\). This formula helps us capture the line of symmetry of the parabola.

In our exercise, with \(f(x) = 5 - 4x - x^2\), we have \(a = -1\) and \(b = -4\). Thus, the x-coordinate of the vertex is \(-\frac{-4}{2(-1)} = -2\). By substituting \(x = -2\) back into the function, we determine the y-coordinate to be \(7\). Therefore, the vertex is \((-2, 7)\).

It's essential to know that the vertex \((h, k)\) informs us about the parabola's position and symmetry. For a downward-opening parabola like ours (since \(a < 0\)), the vertex represents the maximum value the function can achieve.

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. These points are found by setting the function equal to zero and solving for \(x\). The solutions to this equation are the function's x-intercepts.

For the function \(f(x) = 5 - 4x - x^2\), setting it to zero gives us: \(0 = -x^2 - 4x + 5\). By factoring, we find that \( -(x - 1)(x - 5) = 0 \). Solving these, we get the x-intercepts as \(x = 1\) and \(x = 5\). These intercepts correspond to the points \((1, 0)\) and \((5, 0)\) on the graph.

Understanding x-intercepts is vital because they tell us where the parabola crosses the x-axis. This also helps in sketching the graph, as it defines the points where the graph touches or cuts through the axis.

The range of a quadratic function is the set of possible y-values that the function can take. For parabolas, the range largely depends on the vertex and the parabola's orientation (whether it opens upwards or downwards).

In our quadratic function \(f(x) = 5 - 4x - x^2\), which opens downward (because \(a = -1\)), the highest point on the graph is the vertex at \((-2, 7)\). The y-coordinate of the vertex determines the upper limit of the range. Since the parabola opens downward, the function values will go from 7 down to negative infinity.

Therefore, the range of this function is \((-\infty, 7]\). The notation \([\cdot]\) indicates that 7 is included in the range, while \((\cdot)\) signifies that the function continues indefinitely in the indicated direction.