Determining the Vertex and Intercepts of a Quadratic...

Determining the Vertex and Intercepts of a Quadratic Function.
a. Find the vertex.
b. Find the $y$ -intercept.
c. Find the $x$ -intercept(s), if they exist.
d. Use this information to graph the function.
$$y=-x^{2}-2 x+3$$
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Key Concepts

Graphing Parabolas

Graphing quadratic functions involves plotting key features such as the vertex, y-intercept, and x-intercepts, along with drawing the axis of symmetry. Understanding how the coefficient a affects the direction and 'width' of the parabola also facilitates an accurate sketch of the graph.

X-Intercept(s)

The x-intercepts, or roots, of a quadratic function are the points where the graph crosses the x-axis. These can be found by setting the function equal to zero and solving the resulting quadratic equation, often using methods such as factoring, completing the square, or the quadratic formula.

Y-Intercept

The y-intercept of a quadratic function is the point at which the graph crosses the y-axis. This can be found by evaluating the function at x = 0, which directly yields the constant term c in the standard form of the quadratic.

Vertex

The vertex of a quadratic function is the point where the parabola attains its maximum or minimum value. It can be found using the formula (-b/(2a), f(-b/(2a))) from the standard form, or by rewriting the function in vertex form. The vertex represents the axis of symmetry of the parabola.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in standard form as y = ax² + bx + c, where a, b, and c are constants and a is not zero. They are characterized by parabolic graphs that open either upward or downward depending on the sign of a.

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