# Objectives: 1. To identify quadratic functions and graphs 2. To model data with quadratic functions.

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Objectives: 1. To identify quadratic functions and graphs 2. To model data with quadratic functions

 A function that can be written in the standard form :, where  The domain of a quadratic functions is all real numbers Standard form of a Quadratic function: quadratic linear constant term term term

-The graph of a quadratic function is a parabola -The axis of symmetry is the line that divides a parabola into two parts that are mirror images -The vertex of a parabola is the point at which the parabola intersects the axis of symmetry -The y-value of the vertex of a parabola represents the maximum or minimum value of the function -The axis of symmetry of the graph of a quadratic function is always a vertical line defined by the x-coordinate of the vertex

Given three noncollinear points, you can find a quadratic function whose graph passes through those points. Example: Find a quadratic function that includes the values in the table: X Y 2 3 3 13 4 29 Solution: Substitute the values of x and y into The result is a system of 3 linear equations : Set up in a matrix to solve this system: The solution is a = 3, b= -5 and c = 1. Substitute these into the standard form to get:

Find a quadratic function with a graph that includes (1,0), (2,-3) and (3,-10) Substitute the values of x and y into: Set this system up in a matrix and use the calculator to solve. The solution is a = -2, b = 3, and c = -1 So the quadratic equation is:

We can also use the calculator to enter data into lists in the calculator and come up with an equation to model the data. Have to enter x-values and y-values into lists on the calculator: and How to: Hit STAT button Then hit enter to EDIT the two lists. Enter x-values in and enter y-values in To plot data: Hit 2 nd, y = Will say STAT PLOTS Hit enter for Plot 1 Cursor will be on to word “On” ….hit enter Arrow down to type and hit enter on first type of graph Arrow down to Xlist and make sure it has listed Arrow down to Ylist and make sure it has listed Arrow down to Mark and choose first one listed Then hit 2 nd quit

Then hit graph and the data points will be graphed Once the data is in, we can use the calculator to get an approximate equation to model the data How to: Hit the STAT button Arrow right once Then either hit the number 5 or arrow down to QuadReg QuadReg will appear by itself on the screen Hit enter The calculator will then give you the coefficients for the variables and the constant for the quadratic equation. Write these numbers down and use them to make an equation in the y = editor. After entering this into the y = editor, hit graph again, and you will see the graph going through the previously plotted data points Do example 4 on page 240