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Quadratic functions are a fundamental concept in algebra, characterized by an equation of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The graph of a quadratic function is called a parabola.

When graphing a quadratic function such as \( f(x) = x^2 \), we start by determining the shape of the graph. Since the coefficient of \( x^2 \) is positive, the parabola opens upwards. Key points to identify include the vertex of the parabola, which for the function \( f(x) = x^2 \) is at the origin (0,0), and the axis of symmetry, which, in this case, is the y-axis.

The parabola will be symmetric about the axis of symmetry and the vertex is the lowest point on the graph for an upward-opening parabola. By plotting additional points on the graph such as (1,1) and (-1,1), we get a clearer picture of its symmetrical shape. Quadratic functions exhibit this symmetry and U-shape, which are hallmarks of their graphs.

Linear functions are another cornerstone of algebraic graphing, generally represented by an equation of the form \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. These functions create straight lines when graphed on a coordinate plane.

For the linear function given in our exercise, \( g(x) = -2x \), the graph will be a straight line without a y-intercept term (\( b = 0 \)), meaning it crosses the origin. The slope \( m \) is negative, so the line will decrease, or go down and to the right, as \( x \) increases. This linear equation explains that for every one unit you move to the right along the x-axis, you move two units down along the y-axis.

To graph a linear equation like \( g(x) = -2x \), simply plot two points that satisfy the equation and draw a line through them. For example, using (0,0) and (1,-2) gives us enough information to extend the line across the graph. The simplicity and predictability of linear functions make them indispensable for understanding changes between variables.

Function addition involves creating a new function by adding two functions together. The new function, denoted \((f+g)(x)\), is formed by adding the values of \(f(x)\) and \(g(x)\) for any given \(x\).

In our exercise, we are asked to graph \((f+g)(x)\), where \(f(x) = x^2\) and \(g(x) = -2x\). To do this, we add the two functions together to get \((f+g)(x) = x^2 - 2x\). Like \( f(x) \), this combined function is also quadratic due to the \( x^2 \) term, but the \( -2x \) term will affect its shape and position on the graph.

Graphing \( (f+g)(x) = x^2 - 2x \) reveals a parabola shifted from the original \( f(x) \). It is essential to find the vertex of this new parabola, which can be done by completing the square or using the vertex formula \( (-b/(2a), f(-b/(2a))) \). Plotting points, like we did for \( f(x) \) and \( g(x) \), provides more accuracy for the graph. By understanding and practicing function addition, students can tackle more complex graphing problems that involve the merging of multiple function types.