eMathLab: FAQs

eMathLab: Frequently Asked Questions

How do I re-write a quadratic function in standard form?

How Do I re-write a quadratic function in standard form?

A quadratic function can be expressed in the form:

But in its standard form, that is:

we can tell a lot more about it, that is, what its graph looks like. We can tell whether its legs go up or down, depending on whether a is positive (up) or negative (down). We can tell where its vertex is, because its vertex is at the point (h,k). We can also tell whether its legs are "skooshed" together (if the magnitude of a is greater than 1), or spread farther apart (if the magnitude of a is less than 1).

So, the re-writing into standard form is a worthwhile thing to do when we want to get a handle on what the function "looks like."

When we are asked to take a function definition and re-write it in another form, this means that we need to end up with an expression that has the same value as the one we started with.

Here is how we do this, using as our specific example:

Step for

Result of taking step on

If the coefficient of the x²term is not 1, we must factor it out of both the x² term and the x term; leave a blank for the new constant inside the parentheses; leave the old constant just the way it was (NOT in the parentheses); and put the opposite of the coefficient of the x² term times blank.
Now, in the blanks, we need to put the square of half of the linear coefficient (the multiplier of the x term). In our example, the linear coefficient is -6, so we need to put
Now the trinomial in the parentheses is a perfect square. Write it as its square root squared. Since half the linear coefficient is -3, the square root of the trinomial is x - 3. Also, combine the two constant terms.

Notice now that

so we see that we have in fact ended up with the same value that we started with, only written in the form we wished it to be in. Now we know that this parabola's legs open down, are "skooshed" together, and its vertex is at the point (3,23).

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How can I combine fractions that have unlike denominators ?

First of all, we construct the least common denominator of all the fractions to be combined (added/subtracted), and then, for each fraction, we multiply its numerator by factors of the LCD that aren't in its denominator.

We will write down each step, and then on the right you will see the results of taking that step on our example problem:

Step	Result of Taking Step
1. Prime factor each of the denominators
2. Put all the prime factors of the first denominator under the long bar of the result fraction.
3. For each fraction, put under the long bar, as factors, any of its prime factors that are not already there (including repeats). The LCD is now under the long bar, in factored form, as we want it.
4. Now we start to work on the result fraction's numerator. Above the long bar, put each fraction's numerator, a set of empty parentheses, and the sign following
5. For each fraction, put in its numerator's parentheses the factors that are under the long bar but are not in that fraction's own denominator.
6. In the result fraction's numerator, do first the indicated multiplication, then the indicated addition/subtraction (combining terms). Note: Do *NOT* touch anything in the result fraction's denominator during this step.

7. Factor the result fraction's numerator, if possible. Remember, its denominator is *already* in prime factored form. Then, *and only then*, remove any factors that are common to numerator and denominator.

OK, now that we've got the idea, let's try applying the steps to a string of fractions that have algebraic expressions in their denominators and/or numerators. Remember, the process remains exactly the same as it was in the Example above.

Step	Result of Taking Step
1. Prime factor each of the denominators
2. Put all the prime factors of the first denominator under the long bar of the result fraction.
3. For each fraction, put under the long bar, as factors, any of its prime factors that are not already there (including repeats). The LCD is now under the long bar, in factored form, as we want it.
4. Above the long bar, put each fraction's numerator, a set of empty parentheses, and the sign following
5. For each fraction, put in its numerator's parentheses the factors that are under the long bar but are not in that fraction's own denominator.
6. In the result fraction's numerator, do first the indicated multiplication, then the indicated addition/subtraction (combining terms). Note: Do *NOT* touch anything in the result fraction's denominator during this step.

7. Factor the result fraction's numerator, if possible. Remember, its denominator is *already* in prime factored form. Then, *and only then*, remove any factors that are common to numerator and denominator.