ANSWERS
(a) y-intercept: y = 0
x-intercepts: x = 0 and x = -2
(b) f(x) = |x|
(c) Horizontal translation 1 unit left, vertical stretch by a factor of 3 and vertical translation 3 units down
(d) See explanation
(e) Domain: all real values
Range: y ≥ -3
EXPLANATION
(a) The intercepts with the axis are the values where the graph crosses each axis. The y-intercept occurs when x = 0,
[tex]f(0)=3|0+1|-3=0[/tex]And the x-intercept occurs when y = 0. In this case, one of the x-intercepts occurs at the same point as the y-intercept,
[tex]0=3|x+1|-3[/tex]Solve for x. Add 3 to both sides,
[tex]\begin{gathered} 0+3=3|x+1|-3+3 \\ 3=3|x+1| \end{gathered}[/tex]Divide both sides by 3,
[tex]\begin{gathered} \frac{3}{3}=\frac{3|x+1|}{3} \\ 1=|x+1| \end{gathered}[/tex]When we eliminate the absolute value, we have two results: one positive and one negative,
[tex]\begin{gathered} -1=x+1\Rightarrow x=-2 \\ 1=x+1\Rightarrow x=0 \end{gathered}[/tex]Hence, the x-intercepts are x = -2 and x = 0
(b) As we can see, this is an absolute value function. Therefore, the function we'd use to graph it is the absolute value of x,
[tex]f(x)=|x|[/tex](c) The first transformation is given by adding 1 to x. This is a horizontal translation left 1 unit.
Then, we multiply the function by 3, which is a vertical stretch by a factor of 3, and, finally, we subtract 3 from the function, which is a vertical shift 3 units down.
(d) First, we have to graph the parent function and then, apply the transformations from part c.
The graph of the parent function, f(x) = |x| is,
We have to translate it to the left 1 unit, to get the graph of f(x) = |x + 1|,
Next, vertically stretch it by a factor of 3. To do this, we have to multiply the y-coordinates of the points by 3, to obtain the graph of f(x) = 3|x+1|,
And finally, translate it 3 units down,
(e) The domain is the set of all possible x-values for which the function exists. In this case, this is a linear and continuous function, so the domain is all real values.
The range is the set of the y-values the function can take. In this case, the range is the values of y from the vertex and greater, so the range is y ≥ -3
