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Answer:
9) a = ¾, <u>vertex</u>: (-4, 2), <u>Equation</u>: y = ¾|x + 4| + 2
10) a = ¼, <u>vertex</u>: (0, -3), <u>Equation</u>: y = ¼|x - 0| - 3
11) a = -4, <u>vertex</u>: (3, 1), <u>Equation</u>: y = -4|x - 3| + 1
12) a = 1, <u>vertex</u>: (-2, -2), <u>Equation</u>: y = |x + 2| - 2
Step-by-step explanation:
<h3><u>Note:</u></h3>I could <u><em>only</em></u> work on questions 9, 10, 11, 12 in accordance with Brainly's rules. Nevertheless, the techniques demonstrated in this post applies to all of the given problems in your worksheet.
<h2><u>Definitions:</u></h2>The given set of graphs are examples of absolute value functions. The <u>general form</u> 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.
<h2><u>Solutions:</u></h2><h3>Question 9) ⇒ Vertex: (-4, 2)</h3><u>Solve for a:</u>
In order to solve for the value of <em>a</em>, 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 <em>a</em>:
a = ¾
Therefore, given the value of a = ¾, and the vertex, (-4, 2), then the equation of the absolute value function is:
<u>Equation</u>: y = ¾|x + 4| + 2
<h3>Question 10) ⇒ Vertex: (0, -3)</h3>
<u>Solve for a:</u>
In order to solve for the value of <em>a</em>, 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 <em>a</em>:
a = ¼
Therefore, given the value of a = ¼, and the vertex, (0, -3), then the equation of the absolute value function is:
<u>Equation</u>: y = ¼|x - 0| - 3
<h3>Question 11) ⇒ Vertex: (3, 1)</h3>
<u>Solve for a:</u>
In order to solve for the value of <em>a</em>, 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 <em>a</em>:
-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:
<u>Equation</u>: y = -4|x - 3| + 1
<h3>Question 12) ⇒ Vertex: (-2, -2)</h3>
<u>Solve for a:</u>
In order to solve for the value of <em>a</em>, 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 <em>a</em>:
a = 1
Therefore, given the value of a = -1, and the vertex, (-2, -2), then the equation of the absolute value function is:
<u>Equation</u>: y = |x + 2| - 2