(a) graph the quadratic functions on the same rectangular...

(a) graph the quadratic functions on the same rectangular coordinate system and
(b) describe what effect adding a constant, $h$, inside the parentheses has
$f(x)=x^{2}, g(x)=(x-3)^{2},$ and $h(x)=(x+3)^{2}$

Key Concepts

Graphing Quadratic Functions

Graphing quadratic functions involves plotting the parabolic curves based on their standard or vertex form equations. Key features such as the vertex, axis of symmetry, and intercepts are identified and plotted to provide an accurate depiction of the function. When multiple quadratic functions are graphed on the same coordinate system, their relative positions can be compared and analyzed.

Horizontal Translation

Horizontal translation refers to shifting the graph of a function left or right in the coordinate plane. For quadratic functions expressed in the form (x - h)^2, adding a constant inside the parentheses results in a horizontal shift. Specifically, if the function is written as f(x) = (x - h)^2, the graph shifts h units to the right, whereas if it is written as f(x) = (x + h)^2, the graph shifts h units to the left. This transformation does not alter the shape of the parabola, only its horizontal position.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the standard form f(x) = ax^2 + bx + c. Their graphs are parabolas that open upward if the coefficient of x^2 is positive and downward if it is negative. The vertex of the parabola, which represents the minimum or maximum value of the function, plays a key role in understanding the function's behavior.

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