One of the properties of taking an absolute value of a function is that the values are all made positive. The values themselves do not change; only their signs do. In this graph, none of the y-values are negative, so none of them would change. Thus the two graphs should be identical.

First, because the graph consists of pieces that are straight lines, the function must include an absolute value, whose functions usually have a distinctive "V" shape. Thus, we can eliminate f(x) = x² – 4x + 3 from our choices. Furthermore, functions with x² terms are curved parabolas, and do not have straight line segments. This means that f(x) = |x² – 4x| – 3 is not the correct choice. 

Next, let's examine f(x) = |2x – 6|. Because this function consists of an abolute value by itself, its graph will not have any negative values. An absolute value by itself will only yield non-negative numbers. Therefore, because the graph dips below the x-axis (which means f(x) has negative values), f(x) = |2x – 6| cannot be the correct answer. 

Next, we can analyze f(x) = |x – 1| – 2. Let's allow x to equal 1 and see what value we would obtain from f(1). 

f(1) = | 1 – 1 | – 2 = 0 – 2 = –2

However, the graph above shows that f(1) = –4. As a result, f(x) = |x – 1| – 2 cannot be the correct equation for the function. 

By process of elimination, the answer must be f(x) = |2x – 2| – 4. We can verify this by plugging in several values of x into this equation. For example f(1) = |2 – 2| – 4 = –4, which corresponds to the point (1, –4) on the graph above. Likewise, if we plug 3 or –1 into the equation f(x) = |2x – 2| – 4, we obtain zero, meaning that the graph should cross the x-axis at 3 and –1. According to the graph above, this is exactly what happens. 

The answer is f(x) = |2x – 2| – 4.

The graph is a down-opening parabola with a maximum of . Therefore, there are no y values greater than this for this function.

A line has the equation

 where  is the  intercept and  is the slope.

The  intercept can be found by noting the point where the line and the y-axis cross, in this case, at  so .

The slope can be found by selecting two points, for example, the y-intercept and the next point over that crosses an even point, for example, .

Now applying the slope formula,

 which yields .

Therefore the equation of the line becomes:

Graphically, the y-intercept is the point at which the graph touches the y-axis.  Algebraically, it is the value of  when .

Here, we are given the function .  In order to calculate the y-intercept, set  equal to zero and solve for .

So the y-intercept is at .

Graphically, the x-intercept is the point at which the graph touches the x-axis.  Algebraically, it is the value of  for which .

Here, we are given the function .  In order to calculate the x-intercept, set  equal to zero and solve for .

So the x-intercept is at .

A line is defined by any two points on the line.  It is frequently simplest to calculate two points by substituting zero for x and solving for y, and by substituting zero for y and solving for x.

Let .  Then

So our first set of points (which is also the y-intercept) is 

Let .  Then

So our second set of points (which is also the x-intercept) is .

There are four shifts of the graph y = f(x):

y = f(x) + c shifts the graph c units upwards.

y = f(x) – c shifts the graph c units downwards.

y = f(x + c) shifts the graph c units to the left.

y = f(x – c) shifts the graph c units to the right.

The first step in determining which graph is correct is finding the origin of the function. If both x and y are equal to 0, the coordinates of the origin would be . The second step is to determine whether the graph opens up or down. The x and y are both positive, so the parabola will open upwards. The correct graph will look like 

The easiest way to determine which graph belongs to the equation is to find the x-intercept, the slope, and if necessary the y-intercept as well.

This equation is written in  form, where b=y-intercept and m=slope.

Therefore we know that the slope equals , and the y-intercept is .

The x-intercept can be found by substituting 0 for y and solving the equation.

The x-intercept is .

The only graph that meets these standards is