Int. Alg. Notes Section 8.5 Page 1 of 8

Section 8.5: Graphing Quadratic Equations Using Properties

Big Idea: There are formulas that convert between the general form and standard form of a quadratic function.

Big Skill: You should be able to use those formulas to convert between forms so that you can quickly sketch the graph of a quadratic function

Quadratic function in general form:

Quadratic function standard form:

Instead of completing the square every time we are given a quadratic function to graph, we can complete the square on the general form of the quadratic function, and thus get formulas for h and k.

Completing the square on the general form of a quadratic function:

  • Make sure the coefficient of the square term is 1.
  • Identify the coefficient of the linear term; multiply it by ½ and square the result.
  • Add that number to both sides of the equation. Don’t forget the factor of a that distributes onto the number you are adding… Also notice that this step is the same thing as adding and subtracting the same number on the right hand side.
  • Write the resulting perfect square trinomial as the square of the binomial.
  • Compare the completed square to the standard form to identify h and k.

The Vertex of a Parabola

Any quadratic function in general form (a  0) will have its vertex at the point whose coordinates are:

.

Two alternative ways to state the vertex coordinates are using the discriminant:

And by plugging the x-coordinate of the vertex into the function (i.e., since y = f(x) ):

Practice:

  1. Compute the coordinates of the vertex of the parabola specified by the quadratic function.

The x-Intercepts of the Graph of Parabola

The x-intercepts of a graph are the x values where y = 0:

Thus, the x-intercepts of the graph of a parabola are given by the quadratic formula. We can anticipate the number of x-intercepts based on the discriminant:

If the discriminant , then the graph of has two different x-intercepts at .
If the discriminant , then the graph of has one x-intercept, and the vertex of the graph will touch the x-axis at .
If the discriminant , then the graph of has no x-intercepts (the graph does not cross or touch the x-axis).

Practice:

  1. Compute the x-intercepts of the parabola specified by the quadratic function.
  1. Compute the x-intercepts of the parabola specified by the quadratic function.

  1. Compute the x-intercepts of the parabola specified by the quadratic function.

To Graph a Quadratic Function Using Its Properties:

  • Use the formulas and to quickly convert the general form of the quadratic equation, , to the standard form .
  • Graph the standard form using translations.

Practice:

  1. Sketch a graph of using its properties.

  1. Sketch a graph of using its properties.
  1. Sketch a graph of using its properties.

Application of Graphing Quadratic Functions: The vertex of a quadratic function is either the max or min value of the function.

Practice:

  1. A company’s daily revenue R as a function of the price of its product p is given by: . Find the price that maximizes the daily revenue and the maximum revenue.

  1. A farmer has 2000 feet of fencing to enclose a rectangular field. Find the maximum area that can be fenced off and the dimensions of that maximum size field.

Algebra is:

the study of how to perform multi-step arithmetic calculations more efficiently,

and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.