(a) graph each quadratic function by determining whether its...

(a) graph each quadratic function by determining whether its graphs opens up or down and by finding its vertex, axis of symmetry, y-intercepts, and x-intercepts, if any. (b) Determine the domain and the function. (c) Determine where the function is increasing and where it is decreasing. $$ f(x)=3 x^{2}-8 x+2 $$

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial function typically expressed in the standard form f(x) = ax² + bx + c, where a, b, and c are constants. Its graph is a parabola, and the sign of the coefficient a determines whether the parabola opens upward (a > 0) or downward (a < 0). This concept is foundational in understanding how different parameters influence the shape and position of the graph.

Vertex

The vertex of a parabola is its highest or lowest point, depending on whether it opens downward or upward respectively. It can be found using the formula x = -b/(2a) for the x-coordinate, and substituting this back into the function gives the y-coordinate. This point summarizes the optimal value of the quadratic function and is key when graphing.

Axis of Symmetry

The axis of symmetry is the vertical line that splits the parabola into two mirror-image halves. It passes through the vertex and is given by the equation x = -b/(2a). Understanding this concept is crucial for graphing quadratic functions because it reveals the inherent symmetry of the parabola.

Y-intercept

The y-intercept is the point where the graph of the quadratic function crosses the y-axis, found by evaluating the function at x = 0. Since it only depends on the constant term c, this intercept provides a useful starting point for sketching the graph.

X-intercepts

X-intercepts, also known as the roots or zeros of the function, are the points where the graph crosses the x-axis. They are found by solving the quadratic equation ax² + bx + c = 0, often using the quadratic formula. These intercepts indicate where the function changes sign and are important for understanding its behavior relative to the x-axis.

Domain of Quadratic Functions

The domain of a quadratic function includes all real numbers because there are no restrictions on the input values that make the function undefined. This concept is fundamental in understanding that quadratic functions are defined for every real number, making them continuous.

Increasing and Decreasing Intervals

The intervals where a quadratic function is increasing or decreasing are determined by observing the vertex. On a parabola, the function decreases on one side of the vertex and increases on the other. Specifically, if the parabola opens upward, the function decreases up to the vertex and then increases; if it opens downward, it increases up to the vertex and then decreases. This concept is key in calculus and important in understanding the behavior of functions over their domain.

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