Properties of Functions and Graphs - SAT Math

The equation of a parabola can be written in vertex form

where is the vertex and determines if it is a minimum or maximum. If is positive, then it is a minimum; if is negative, then it is a maximum.

In this example, is negative, so the vertex is a maximum.

and

Rewrite the equation by the order of powers.

This is a parabola in standard form:

Determine the values of to the vertex formula.

Since the leading coefficient of the parabola is negative, the parabola will curve downward and will have a maximum point.

Therefore there is a maximum at,

.

To find the domain, find all areas of the number line where the fraction is defined.

because the denominator of a fraction must be nonzero.

Factor by finding two numbers that sum to -2 and multiply to 1. These numbers are -1 and -1.

The domain includes the values that go into a function (the x-values) and the range are the values that come out (the or y-values). A sine curve represent a wave the repeats at a regular frequency. Based upon this graph, the maximum is equal to 1, while the minimum is equal to –1. The x-values span all real numbers, as there is no limit to the input fo a sine function. The domain of the function is all real numbers and the range is .

The easiest way to solve for this kind of simple -intercept is to set equal to . You can then solve for the value in order to find the relevant intercept.

Solve for :

Divide both sides by 40:

The domain is the set of possible value for the variable. We can find the impossible values of by setting the denominator of the fractional function equal to zero, as this would yield an impossible equation.

Now we can solve for .

There is no real value of that will fit this equation; any real value squared will be a positive number.

The radicand is always positive, and is defined for all real values of . This makes the domain of the set of all real numbers.

Using as the input () value for this equation generates an output () value that contradicts the stated condition of .

Therefore is not a valid value for and not in the equation's domain:

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of , there can only be one value of . For the function , there is one value for two possible values. For instance, if , then . But if , as well. This function fails the vertical line test. The other functions listed are a line,, the top half of a right facing parabola, , a cubic equation, , and a semicircle, . These will all pass the vertical line test.

This function is a parabola that has been shifted up five units. The standard parabola has a range that goes from 0 (inclusive) to positive infinity. If the vertex has been moved up by 5, this means that its minimum has been shifted up by five. The first term is inclusive, which means you need a "\[" for the beginning.

Minimum: 5 inclusive, maximum: infinity

Range:

The domain represents the acceptable values for this function. Based on the members of the function, the only limit that you have is the non-allowance of a negative number (because of the square root). The square and the linear terms are fine with any numbers. You cannot have any negative values, otherwise the square root will not be a real number.

Minimum: 0 inclusive, maximum: infinity

Domain:

There are two limitations in the function: the radical and the denominator term. A radical cannot have a negative term, and a denominator cannot be equal to zero. Based on the first restriction (the radical), our term must be greater than or equal to zero. Based on the second restriction (the denominator), our term cannot be equal to 4. Our final answer will be the union of these two sets.

Minimum: 0 (inclusive), maximum: infinity

Exclusion: 4

Domain:

The domain of a function refers to the viable value inputs. Common domain restrictions involve radicals (which cannot be negative) and fractions (which cannot have a zero denominator).

This function does not have any such restrictions; any value of will result in a real number. The domain is thus unlimited, ranging from negative infinity to infinity.

Domain:

This function represents a parabola that has been shifted 15 units to the left and 2 units up from its standard position.

The vertex of a standard parabola is at (0,0). By shifting the graph as described, the new vertex is at (-15,2). The value of the vertex represents the minimum of the range; since the parabola opens upward, the maximum will be infinity. Note that the range is inclusive of 2, so you must use a bracket "\[".

Minimum: 2 (inclusive), maximum: infinity

Range:

The domain of a function refers to the viable value inputs. Common domain restrictions involve radicals (which cannot be negative) and fractions (which cannot have a zero denominator). Both of these restrictions can be found in the given function.

Let's start with the radical, which must be greater than or equal to zero:

Next, we will look at the fraction denominator, which cannot equal zero:

Our final answer will be the union of the two sets.

Minimum: 2 (inclusive), maximum: infinity

Exclusion: 22

Domain:

The range of a function is the group of corresponding values for a given domain ( values). Plug each value into the function to find the range:

The range is .

The radicand within a square root symbol must be nonnegative, so

This happens if and only if , so the domain of is .

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.

Upon doing so, we learn that the -values (domain) are not restricted on either end of the function, creating a domain of negative infinity to postive infinity.

If we plug in -100000 for , we get 100000 for .

If we plug in 100000 for , we get 100000 for .

Additionally, if we plug in any value for , we will see that we always get a real, defined value for .

**Extra Note: Due to the absolute value notation, the negative (-) next to the is not important, in that it will always be made positive by the absolute value, making this function the same as . If the negative (-) was outside of the absolute value, this would flip the function, making all corresponding -values negative. However, this knowledge is most important for range, rather than domain.

For an equation like this, you should use the quadratic formula to solve for the roots. We can easily get our equation into proper form by substituting for :

Recall that the general form of the quadratic formula is:

Based on our equations, the following are your formula values:

Therefore, the quadratic formula will be:

Simplifying, you get:

Using a calculator, you will get:

and

A basic knowledge of absolute value and its functions is valuable for this problem. However, if you do not know what the typical shape of an absoluate value function looks like, one can always plug in values and plot points.

Upon doing so, we learn that the -values (range) never surpass . This is because of the negative that is placed outside of the absolute value function. Meaning, for every value we plug in, we will always get a negative value for , except when .

With this knowledge, we can now confidently state the range as

**Extra note: the negative sign outside of the absolute value is simply a transformation of , reflecting the function about the -axis.

The radicand within a square root symbol must be nonnegative, so

This happens if and only if , so the domain of is .

assumes its greatest value when , which is the point on where is least - this is at .

Similarly, assumes its least value when , which is the point on where is greatest - this is at .

Therefore, the range of is .