9) a = ¾, vertex: (-4, 2), Equation: y = ¾|x + 4| + 2
10) a = ¼, vertex: (0, -3), Equation: y = ¼|x - 0| - 3
11) a = -4, vertex: (3, 1), Equation: y = -4|x - 3| + 1
12) a = 1, vertex: (-2, -2), Equation: y = |x + 2| - 2
Step-by-step explanation:
Note:
I could only work on questions 9, 10, 11, 12 in accordance with 's rules. Nevertheless, the techniques demonstrated in this post applies to all of the given problems in your worksheet.
Definitions:
The given set of graphs are examples of absolute value functions. The general form of absolute value functions is: y = a|x – h| + k, where:
|a| = determines the vertical stretch or compression factor (wideness or narrowness of the graph).
(h, k) = vertex of the function
x = h represents the axis of symmetry.
Solutions:Question 9) ⇒ Vertex: (-4, 2)
Solve for a:
In order to solve for the value of a, choose another point on the graph, (0, 5) and substitute into the general form (equation):
y = a|x – h| + k
5 = a| 0 - (-4)| + 2
5 = a| 0 + 4 | + 2
5 = a|4| + 2
5 = 4a + 2
Subtract 2 from both sides:
5 - 2 = 4a + 2 - 2
3 = 4a
Divide both sides by 4 to solve for a:
a = ¾
Therefore, given the value of a = ¾, and the vertex, (-4, 2), then the equation of the absolute value function is:
Equation: y = ¾|x + 4| + 2
Question 10) ⇒ Vertex: (0, -3)
Solve for a:
In order to solve for the value of a, choose another point on the graph, (4, -2) and substitute into the general form (equation):
y = a|x – h| + k
-2 = a|4 - 0| -3
-2 = a|4| - 3
-2 = 4a - 3
Add 3 to both sides:
-2 + 3 = 4a - 3 + 3
1 = 4a
Divide both sides by 4 to solve for a:
a = ¼
Therefore, given the value of a = ¼, and the vertex, (0, -3), then the equation of the absolute value function is:
Equation: y = ¼|x - 0| - 3
Question 11) ⇒ Vertex: (3, 1)
Solve for a:
In order to solve for the value of a, choose another point on the graph, (4, -3) and substitute into the general form (equation):
y = a|x – h| + k
-3 = a|4 - 3| + 1
-3 = a|1| + 1
-3 = a + 1
Subtract 1 from both sides to isolate a:
-3 - 1 = a + 1 - 1
a = -4
Therefore, given the value of a = -4, and the vertex, (3, 1), then the equation of the absolute value function is:
Equation: y = -4|x - 3| + 1
Question 12) ⇒ Vertex: (-2, -2)
Solve for a:
In order to solve for the value of a, choose another point on the graph, (-4, 0) and substitute into the general form (equation):
y = a|x – h| + k
0 = a|-4 - (-2)| - 2
0 = a|-4 + 2| - 2
0 = a|-2| - 2
0 = 2a - 2
Add 2 to both sides:
0 + 2 = 2a - 2 + 2
2 = 2a
Divide both sides by 2 to solve for a:
a = 1
Therefore, given the value of a = -1, and the vertex, (-2, -2), then the equation of the absolute value function is:
Equation: y = |x + 2| - 2
