Algebra II - Graphing Absolute Value Functions

Axes_1

Refer to the above figure.

Which of the following functions is graphed?

Explanation

Below is the graph of :

Axes_1

The given graph is the graph of reflected in the -axis, then translated up 6 units. This graph is

, where .

The function graphed is therefore

Give the vertex of the graph of the function .

None of the other choices gives the correct response.

Explanation

Let

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates . In terms of ,

or, alternatively written,

The graph of is the same as that of , after it shifts 10 units left ( ), it flips vertically (negative symbol), and it shifts up 10 units (the second ). The flip does not affect the position of the vertex, but the shifts do; the vertex of the graph of is at .

Which of the following absolute value functions is represented by the following graph?

$f(x)=2\left | x-3\right |+1$

$f(x)=2\left | x+3\right |+1$

$f(x)=3\left | x+3\right |+1$

$f(x)=3\left | x-3\right |+1$

The equation cannot be determined from the graph.

Explanation

The equation can be determined from the graph by following the rules of transformations; the base equation is:

$f(x)=\left | x\right |$

The graph of this base equation is:

When we compare our graph to the base equation graph, we see that it has been shifted right 3 units, up 1 unit, and our graph has been stretched vertically by a factor of 2. Following the rules of transformations, the equation for our graph is written as:

$f(x)=2\left | x-3\right |+1$

Screen_shot_2014-12-24_at_3.03.30_pm

What is the equation of the above function?

$y=|\frac{x}{2}+2|-3$

Explanation

The formula of an absolute value function is where m is the slope, a is the horizontal shift and b is the vertical shift. The slope can be found with any two adjacent integer points, e.g. and , and plugging them into the slope formula, $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ , yielding . The vertical and horizontal shifts are determined by where the crux of the absolute value function is. In this case, at , and those are your a and b, respectively.