6-6 Analyzing Graphs of Quadratic Functions Objective

Up until this point, when we graphed a quadratic function we needed to put it in standard form, which looked like: Another form that a quadratic function can be written in is vertex form. This looks like: The vertex form of a quadratic function gives us certain information that makes it very easy to graph the function.

The vertex of the function is the ordered pair (h, k). Note that in vertex form, h is negative. Therefore, when stating the ordered pair of the vertex, h will be the opposite sign of what it is in the function. The axis of symmetry is the line x=h. Remember that when a is positive, the parabola opens upwards, and when a is negative, the parabola opens downwards.

The parent graph of a parabola is the equation: Just like when we examined absolute value functions in Chapter 2, the graph of a parabola will translate based on certain values being added, subtracted, or multiplied to our parent graph.

The k value on the outside of the parenthesis: – If k is positive, the graph moves up that many spaces. – If k is negative, the graph moves down that many spaces. The h value on the inside of the parenthesis: The a value being multiplied to the function:

Example 1: For each quadratic function, identify the vertex, axis of symmetry, and direction of opening. Then, graph the function. 1)

2)

Try these. 3)4)

When the function is not already in vertex form, we have to get it in vertex form by completing the square. When completing the square, we only work one side of the equation. Therefore, instead of adding the same constant to both sides of the equation, we must add and subtract that constant from the same side of the equation. This will ensure our equation stays balanced. Let’s try…

Example 2: Write each quadratic function in vertex form. Then, identify the vertex, axis of symmetry, and direction of opening. 1) 2)

Try this. 3)

4)5)

Try this. 6)

When given the vertex of a parabola and point a parabola passes through, we can write the equation for the parabola. The process is similar to writing a linear equation when given its slope and a point the line passes through. To write the equation for a parabola, substitute the vertex and ordered pair into the vertex form equation, and then solve for a. Then go back to the vertex form equation, and substitute in a, h, and k.

Example 3: Write an equation for the parabola with the given vertex that passes through the given point. 1) Vertex (-3, 6); passes through (-5, 2)

2) Vertex (2, 0); passes through (1, 4)

Try this. 3) Vertex (1, 3); passes through (-2, -15)