Answer:
Step-by-step explanation:
Definitions:
The absolute value equation in vertex form, given by: y = a|x – h| + k
where:
(h, k) = vertex
a = determines whether the graph opens up or down. If a > 1, the graph opens up. If a < 1, the graph opens down.
h = determines how far left or right the graph is translated.
k = determines how far up or down the graph is translated.
The absolute value inequalities are graphed the same way as graphing the absolute value equations.
Given the absolute value inequality, y ≤ - ⅔ |x - 3| + 4
where:
a = - ⅔
h = 3
k = 4
In reference to the definitions provided in this post, the vertex is (h, k). In the given inequality statement, the vertex occurs at point (3, 4). Given the value of a = - ⅔, it means that the graph opens down. To find other points on the graph, we can solve for the intercepts.
Finding the Intercepts to Graph:
The y-intercept is the point on the graph where it crosses the y-axis. It is also the value of y when x = 0.
To solve for the y-intercept, set x = 0:
y = - ⅔ |0 - 3| + 4
y = - ⅔ |- 3| + 4
y = - 2 + 4
y = 2 ⇒ y-intercept: (0, 2).
The x-intercept is the point on the graph where it crosses the x-axis. It is also the value of x when y = 0.
To solve for the x-intercept, set y = 0:
0 = - ⅔ |x - 3| + 4
0 - 4 = - ⅔ |x - 3| + 4 - 4
-4 = - ⅔ |x - 3|
-4 (3) = (- ⅔ |x - 3| ) (3)
-12 = -2 |x - 3|
Divide both sides by -2:
[tex]\frac{-12}{-2} = \frac{-2 |x - 3| }{-2}[/tex]
|x - 3| = 6
Apply absolute rule:
x - 3 = 6 or x - 3 = - 6
x - 3 + 3 = 6 + 3 or x - 3 + 3 = - 6 + 3
x = 9 or x = -3
Therefore, the x-intercepts are: (-3, 0) and (9, 0).
Graphing steps:
You now have the following points to plot on the graph:
vertex: (3, 4)
y-intercept: (0, 2)
x-intercepts: (-3, 0) and (9, 0).
You could easily connect these points with lines. In graphing the boundary line, you will use a solid line due to the inequality symbol, "≤."
The last step involves shading the appropriate half-plane region. In order to do this, choose a test point that is not on the boundary lines. We can choose point, (0, 0). Substitute these coordinates into the absolute value inequality. If it provides a true statement, then you'll shade the region that contains that test point.
Test point: (0, 0)
y ≤ - ⅔ |x - 3| + 4
0 ≤ - ⅔ |0 - 3| + 4
0 ≤ [tex]-\frac{2|0 - 3|}{3} + 4[/tex]
0 ≤ [tex]-\frac{|- 6|}{3}[/tex] + 4
0 ≤ - 2 + 4
0 ≤ 2 (True statement. Shade the region where (0, 0) is included).
Attached is a screenshot of the graphed absolute value inequality.
