Answer:
- C. y=5√(x-8) +11
- A. y=√(x+7)
- A. (-12, 0)
- B. x ≥ -4
- D. y=-5∛x +15
Step-by-step explanation:
These questions have to do with translation and scaling of functions.
The functions being translated and scaled are the square root and cube root functions, so you are expected to have some idea what those look like. In particular, you need to know the "end point" of the square root function, and the point of inflection of the cube root function. (Both are the point (0, 0).) You also need to know that, generally, these functions increase from left to right.
The transformations of concern are horizontal and vertical translation, and vertical scaling:
f(x) ⇒ f(x) + k . . . . . the function is translated upward k units
f(x) ⇒ f(x -h) . . . . . . the function is translated to the right h units. (If h is negative, so it looks like x+a, then the function is translated <em>left</em> a units.)
f(x) ⇒ a·f(x) . . . . . . . the function is scaled vertically by a factor of "a". Among other things, this multiplies the slope of the function by "a".
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<h2>1.</h2>
The square root function has its endpoint moved from (0, 0) to (8, 11), so we have h=8, k=11, which is a combination of horizontal and vertical translation. Most of the answer choices are also scaled vertically by a factor of 5 (which does not move the end point). So, the square root function becomes ...
y = 5√x . . . . scaled vertically by a factor of 5
y = 5√x + 11 . . . . also translated up by 11 units (k=11)
y = 5√(x -8) + 11 . . . . also translated right by 8 units (h=8)
This translated and scaled function matches selection C.
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<h3>2.</h3>
The function with the least steep graph will be the one with the vertical scaling of the smallest magnitude. The vertical scale factors in the answer choices are 1, -12, 3, -5. The one of these with the smallest magnitude is 1, corresponding to answer selection A.
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<h2>3.</h2>
The graph of
y = -2∛(x+12)
is a graph of the cube root function scaled vertically by a factor of -2 and translated with (h, k) = (-12, 0). Since the original point of inflection is at (0, 0), the translated point will be at (-12, 0), matching selection A.
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<h2>4.</h2>
The end point of the square root function marks the lower end of the function's domain. That is, the straight square root function has a domain of [0, ∞). When the endpoint is translated to the left from 0 to -4, the domain of the translated function becomes [-4, ∞). That is, x ≥ -4, corresponding to selection B.
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<h2>5.</h2>
The square root function has a range of [0, ∞), while the cube root function has a range of "all real numbers", or (-∞, ∞). The cube root function normally increases from left to right, but if it is vertically scaled by a negative number, it will decrease from left to right. The only selection that is a negative scaling of the cube root function is selection D.
