Quadratic Functions

Quadratic Functions
Quadratic functions are any functions that may be written in the form y = ax2 + bx + c where a, b, and c are real coefficients and a ≠ 0. For example, y = 2x2 is a quadratic function since we have the x-squared term. y = x2 – 1/x + 1 would not be a quadratic function because the 1/x term is equal to x-1 which does not fit the form. The graph of a quadratic function is called a parabola. The Vertex Of The Graph Of A Quadratic Function The vertex of the graph of a quadratic function is defined as the point where the graph changes from increasing to decreasing or changes from decreasing to increasing. Several graphs are shown below along with location of each vertex. Note that the vertex of y=x2 will be (0,0).

Just Remember, The Vertex is the highest or lowest point of the graph. Graphing y=x2 and Shifts of y=x2 We can obtain an equation for the graph of any quadratic function we want by applying function graph shift rules to the graph of y=x2. For example, if we want a graph that is a vertical stretch of y=x2 by a factor of two, is shifted right 4 units, and opens downward instead of upward, we use the formula y = -2(x - 4)2. The graphs of both are shown below.

Note that the vertex has been shifted from (0,0) right 4 units. We use the vertex as our reference point for this new parabola with vertex shifted 4 units right to (4,0).

From MathMotivation.com – Permission Granted For Use and Modification For Non-Profit Purposes

Example: Use function shift rules to predict what the graph of y = 3(x – 4)2 + 1 will look like. Also, indicate the location of the vertex. In this case, we are vertically stretching y = x2 by a factor of 3 (Vertical Stretch Rule) and then shifting to the right 4 and up 1 (Function Shift Rules for Shifting Right and Up). The vertex is shifted 4 to the right and up 1 from (0,0) to (4,1).

Standard Form of a Quadratic Functions The Standard Form of a quadratic function is y = a(x – h)2 + k If a quadratic function is...