Algebra Tutorials!
Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# What the Factored Form of a Quadratic can tell you about the graph

Like standard and vertex forms, the factored form of a quadratic function:

y = a Â· (x - c) Â· (x - d)

can also tell you whether the given quadratic has a graph that â€œsmilesâ€ or â€œfrowns.â€ As with the standard and vertex forms the key again is the sign (+ or -) of the number a. If a is positive the graph â€œsmilesâ€ and if a is negative the graph â€œfrowns.â€ Sometimes you will see an example of a factored form that does not appear to have a value of a, such as:

y = (x -1) Â· (x + 2).

In this case, the value of a is equal to one (which is positive) and the graph of the quadratic will smile.

The factored form of a quadratic equation also tells you where the x-intercepts (sometimes called the roots or zeros of the quadratic function are located). The xintercepts of the equation are the x-values that will make y = 0.

For example, the x-intercepts of the quadratic:

y = (x -1) Â· (x + 2)

are x = 1 and x = -2 as plugging either of these two values into the quadratic equation will make y equal to zero.

Example

Figure 1 shows the graph of a quadratic function. Find the equation of this quadratic and express your answer in both factored and standard forms.

Figure 1: Find the formula of this quadratic function.

Solution

Figure 1 clearly shows both the of the x-intercepts of the quadratic function so it will be easiest to find the factored form of the quadratic first and then convert this to standard form by FOILing.

The x-intercepts of the quadratic shown in Figure 4 are located at x = 0 and x = 4. This means that the factored form of the quadratic function must look something like this:

y = a Â· (x - 0) Â· (x - 4) = a Â· x Â· (x - 4) .

The factored form must have a factor of (x - 0), or more simply a factor of just x, to ensure that when you plug in x = 0 the value of y will be equal to zero. The factored form must also have a factor of (x - 4) to ensure that when you plug in x = 4 the value of y will be equal to zero.

To determine the numerical value of a you can plug in the x- and y-coordinates of any other point on the quadratic graph (i.e. any point other than one of the x-intercepts) and solve for a. Figure 1 shows that the point (2, 2) lies on the graph, so you can plug in x = 2 and y = 2 into the factored form. Doing this:

2 = a Â· 2 Â· (2 - 4)

2 = a Â· (-4)

So, the equation of the quadratic function from Figure 1 (written in factored form) is:

To convert this equation to standard form, you can expand by FOILing and then simplify (if necessary). Doing this:

(Expand by FOILing)

(Multiply through by - beware of â€œ-â€ signs)

The formula for the quadratic shown in Figure 1 (expressed in standard form) is: