Graph each function using end behavior, intercepts, and...

Graph each function using end behavior, intercepts, and completing the square to write the function in shifted form. Clearly state the transformations used to obtain the graph, and label the vertex and all intercepts (if they exist). Use the quadratic formula to find the $x$ intercepts. $$g(x)=x^{2}-6 x-7$$

Key Concepts

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form ax² + bx + c = 0. It, derived from completing the square, provides the x-intercepts (or roots) of the quadratic function regardless of whether the solutions are real or complex, thereby being an essential tool in algebra.

Intercepts

Intercepts are the points at which the graph crosses the coordinate axes. The y-intercept is obtained by setting x = 0 in the function, while the x-intercepts are found by solving the equation f(x) = 0. These points are crucial for sketching the graph since they indicate where the function intersects the axes.

Vertex Form and Transformations

Writing a quadratic function in vertex form (shifted form) directly displays the vertex of the parabola and the transformations it underwent compared to the parent function f(x) = x². This form highlights the horizontal and vertical shifts, making it easier to graph the function and understand its structure.

End Behavior

The end behavior of a quadratic function refers to the values of the function as x approaches positive or negative infinity. For instance, if the leading coefficient is positive, the function’s output increases without bound as x goes to both positive and negative infinity, thereby shaping the general outline of the parabola.

Quadratic Functions

Quadratic functions are polynomial expressions of degree two, generally written in the form ax² + bx + c. The graphs of quadratic functions are parabolic, opening upward when a > 0 and downward when a < 0. Understanding quadratic functions is essential as they model various natural phenomena and are fundamental in algebra and calculus.

Completing the Square

Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial plus or minus a constant. This method is particularly useful for rewriting the quadratic equation into vertex form, which directly reveals the vertex and simplifies the graphical representation by showing horizontal and vertical shifts.