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.
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)= |3/4(x-2)|-7
⇓
g(x)=1 |1/43(x- 2)|+( -7)
We see above that a = 1, b = 43, h = 2, and k = - 7. Then, the vertex of g(x) is at (2,-7), which means that the parent function is translated 2 units right and 7 units down.
( 0, 0) → ( 2, -7)
Since |b|= 43, then g(x), in addition of being translation, is also a horizontal stretch of the parent function by a factor of 43. The y-coordinate of each point on the graph of the parent function will be shifted 7 units down, and the x-coordinate will be stretched by a factor of 43 and then moved 2 units to the right. Let's consider the points (-6,6) and (6,6).
(-6,6) &→ (4/3(-6)+ 2,|6|- 7) → (-6,-1)
(6,6) &→ (4/3(6)+ 2,|6|- 7) → (10,-1)
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
