In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = - (x + 1)^2 + 4

Key Concepts

X-Intercept

The x-intercept of a graph is the point where the graph intersects the x-axis. This occurs when the value of y is zero. To find the x-intercept, you set the equation equal to zero and solve for x. In the context of the given equation, identifying the x-intercept is crucial for understanding the graph's behavior and its relationship to the equation.

Quadratic Functions

The equation provided, y = - (x + 1)^2 + 4, is a quadratic function, which is characterized by its parabolic shape. Quadratic functions can open upwards or downwards depending on the sign of the leading coefficient. In this case, the negative sign indicates that the parabola opens downwards, affecting the location of the vertex and the x-intercepts.

Vertex Form of a Quadratic

The vertex form of a quadratic function is expressed as y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. The given equation can be rewritten to identify its vertex, which helps in sketching the graph and understanding its maximum or minimum point. The vertex is crucial for determining the range of the function and the position of the x-intercepts.