What is the standard form of a function?

Question

This question is important for understanding the key features of functions.

Answer 1

The standard form of a function depends on what type of function it is. For example, the standard form of a linear function is Ax + By = C and the standard form of a quadratic function is Ax squared plus Bx plus C equals 0. In both functions the letters A and B refer to coefficients and C is a constant.

Consider the function 3x + 12y + 13x - 22 = 2. Top put it in standard form. we first need to combine the like terms. 3x can be combined with 13x and the two constants can also be combined. The result is 16x + 12y = 24. But that's still not it's standard form because the numbers can all be divided by 4. After factoring out 4 from each term, we get 4x + 3y = 6

We can see on a graphing calculator that changing the form of the function does not change its graph. However, the standard form gives us information about key features of the function that the first equation does not. For example,If we divide the constant 6 by the y coefficient of 3, that gives us 2, which is the y intercept. Dividing 6 by the x coefficient of 4 gives us 3/2 three halves or 1.5, which is the x intercept.

The standard form is also useful for solving systems of linear equations. Consider the function -2x + 3y = -3. If we want to determine where these two lines intersect, we use either the substitution or the elimination method. Because the coefficients of the y variables are both 3, subtracting one equation from the other is the simplest way to eliminate one of the variables.

6x + 0y = 9

x = 9/6 = 3/2 = 1.5

We see that the x coordinate will equal 1.5. In the next step we replace x with 1.5 in either equation and the result will be y equals 0. The intersection of the two lines is the point (1.5, 0). It's the x intercept of both functions.

In order to solve other types of linear function problems, you need to know two other forms: the slope intercept form and the point slope form. We often convert a linear function in the standard form to the slope intercept form to determine the function's slope. We refer to the slope intercept form as y = mx + b, where m is the slope and b is the y intercept. The letter m also represents the slope in the point slope form. We use this form when we know the slope of the line and one point. We can then determine all the other points in the function.

Note that our first function 4x + 3y = 6 forms a line that falls as it moves from left to right. That indicates the slope is negative. By changing the standard form so that y is by itself on one side of the equation, we have put the function in the slope intercept form. We can now see that the slope is minus 4 thirds. y = -4x/3 + 2

Entering this form of the function in our graphing calculator produces the same line as the standard form.

Knowing how to put a function into the right form is the key to solving many algebra problems.