Question

Please work the problems on a separate piece of paper, take a picture, and upload.

1. Graph the equation.


2. Graph the equation and describe the transformation from the parent function f left parenthesis x right parenthesis space equals space open vertical bar x close vertical bar.


3. Without graphing, identify the vertex, axis of symmetry, and transformations form the parent function


4. Write an absolute value equation for the graph.


5. Graph the equation.

Answer 1

1.

A picture is shown of the graph of
`y=|x|`. To graph
`y=|x|-3`, we use the translation property of graphs. If
`y=f(x)` is a function given, then
`y=f(x)+c` (c being a constant) will shift the curve c units up. Similarly if c is negative, it is going to shift the curve downwards c units. Hence, the graph of the function
`y=|x|-3` is
`y=|x|` shifted 3 units down. Attached picture shows the new graph.

2.

Attached picture shows the parent function
`f(x)=|x|` in blue and the new transformed function
`g(x)=-(3)/(4) |x|` in red.

• The negative sign in front of
  `|x|` reflects the line about the x axis.
• The
  `(3)/(4)` in front of the function makes the corresponding y values for all x less (
  `(3)/(4)` of all values in the original).

3.

For the given function
`y=4-|x+2|`, we write a general form of the absolute value function and identify the constants.

General form,

`y=a|x+[tex](-b,c)`b|+c[/tex]

• If a is positive, graph opens upward and opens downward when a is negative.
• +b translates the original graph (
  `y=|x|`) b units left and -b translates the original b units right
• +c translates the original graph c units up and -c translates it c units down
• Axis of symmetry is given as
  `x=-b`
• Vertex is given as (-b,c)

As noted, we can now say that the vertex is
`(-2,4)` and axis of symmetry as
`x=-2`.

As for transformations (comparing with parent function), we can note the following:

• reflected about x axis, opens downward
• translated 2 units left
• translated 4 units up

4.

From the vertex shown be know that this graph is translated 2 units left and 6 units down. So by using the properties noted above, we can write
`y=|x+2|-6`. But the graph, compared to the parent function
`|x|` ) is a little wider. So there is a value for a, the coefficient before the absolute value sign. To figure it out, we can take any 2 points on the graph and solve for a. Let us take a random point
`(0,-5)`.

`y=a|x+2|-6`, pluggin 0 in x and -5 in y gives us,

`-5=a|0+2|-6\\-5=2a-6\\1=2a\\a=(1)/(2)`.

So our final equation is
`y=(1)/(2)|x+2|-6`

5.

From the equation we can make out that the graph is shifted 2 units left and 3 units down compared to the parent
`|x|`. Also, the 2 in front tells us all y values are twice of the original y values of the parent function for all values of x. Attached graph is shown.