The given function is in the form g(x)=a | 1b(x-h)|+k, where (h,k) is the vertex, a is a parameter for vertical stretch/compression, and b is a parameter for horizontal stretch/compression.
Graph:
Domain: All real numbers Range: y≥ 7
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The given function is in the form of g(x)=a | 1b(x-h)|+k, where (h,k) is the vertex, a is a parameter for vertical stretch/compression, and b is a parameter for horizontal stretch/compression.
g(x)=4/3 |(x-5)|+7
⇓
g(x)=4/3 |1/1(x- 5)|+ 7
We see above that a = 43, b = 1, h = 5, and k = 7. Then, the vertex of g(x) is at (5,7), which means that the parent function is translated 5 units right and 7 units up.
( 0, 0) → ( 5, 7)
Since a>1, then g(x), in addition of being a translation, is also a vertical stretch of the parent function by a factor of 43. The x-coordinate of each point on the graph of the parent function will be shifted 5 units to the right, and the y-coordinate will be stretched by a factor of 43 and then moved up 7 units. Let's consider the points (-3,3) and (3,3).
(-3,3) & → (-3+ 5,4/3|3|+ 7) → (2,11)
(3,3) & → ( 3+ 5,4/3|3|+ 7) → (8,11)
Now, we will plot the vertex and the above points, and graph g(x). Recall that the graph of an absolute value function has a V-shape!
We see above that there are no restrictions for the values that the x-variable can take. Moreover, we also see that the y-variable takes values that are greater than or equal to 7. We will use this information to write the domain and range of the function.
Domain:& all real numbers
Range:& y≥ 7
