Find the vertex of the graph of each quadratic function....

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find the y-intercept, approximate the $x$ -intercepts to one decimal place, and graph the function.
$$
f(x)=x^{2}+10 x+15
$$

Key Concepts

X-intercepts

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis, found by setting f(x) = 0 and solving the resulting quadratic equation. These intercepts can be determined using methods such as factoring, completing the square, or applying the quadratic formula. X-intercepts reveal the roots of the equation and are key to understanding the function's behavior across the x-axis.

Graphing Quadratic Functions

Graphing a quadratic function involves plotting key features such as the vertex, axis of symmetry, x-intercepts, and y-intercept. Once these components are identified, they can be connected with a smooth curve to form a parabola. A clear understanding of these elements allows for an accurate sketch of the function and a visual representation of its properties.

Y-intercept

The y-intercept of a quadratic function is the point at which the graph crosses the y-axis. It is found by evaluating the function at x = 0, which yields f(0) = c. The y-intercept provides a starting point for graphing the quadratic function and is essential for sketching its overall shape.

Quadratic Function

A quadratic function is a function of the form f(x) = ax² + bx + c, where a, b, and c are constants and a is not zero. This type of function generates a parabola when graphed, and its properties such as the vertex, axis of symmetry, and intercepts play a key role in understanding its behavior and graph.

Vertex

The vertex of a quadratic function is the highest or lowest point on its graph, depending on whether the parabola opens downward or upward. It can be found using the formula x = -b/(2a) for the x-coordinate, and then substituting back into the function to obtain the y-coordinate. The vertex is important as it provides critical information about the maximum or minimum value of the function.

Direction of Opening

The direction in which a parabola opens is determined by the sign of the coefficient a in the quadratic function. If a is positive, the parabola opens upward, making the vertex the minimum point. If a is negative, the parabola opens downward, making the vertex the maximum point. This property helps in understanding the overall behavior of the graph.

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