Question

Make a sketch of the graph the absolute value equation.Be sure to label the increments on your x and y axis the vertex as well as the intercepts.

Answer 1

Solution:

Let the following function:

$f(x)=\text{ }\frac{1}{2}\text{|x+4|-3}$

The graph of this function can be obtained by applying the respective function transformations to the absolute value function y = |x|. In this case, horizontal and vertical translations and vertical shortening were used on absolute value function y = |x|.

According to the function, the vertex can be obtained by solving the following equation:

$\text{|x+4|}=0$

solving for x, we get:

$x\text{ = -4}$

replacing this value into the function f(x) =y, we obtain:

$y\text{ = }-3$

so that, the vertex of this function is on the point:

$(x,y)=(-4,-3)$

Now, to find the x-intercept, we set the equation of the function equal to 0 and then solve for x:

$0=\text{ }\frac{1}{2}\text{|x+4|-3}$

solving for x, we get two solutions:

$x=\text{ -10}$

and

$x=\text{ }2$

so that, the x-intercepts are the points:

$(x,y)=(-10,0)$

and

$(x,y)=(2,0)$

On the other hand, to find the y-intercept, we can evaluate the function f(x) at x=0, and then, we can solve for y:

$f(0)=\text{ }\frac{1}{2}\text{|0+4|-3}=\frac{1}{2}\text{|4|}-3\text{ = 2-3 = -1}$

Then, the y-intercept is on the point:

$(x,y)=(0,-1)$

So that, we can conclude that the correct answer is: