1.
Item 1. Which transformation is represented in this graph?
Solution:The two parabolas have the same shape and size, but they are mirror images across the \(y\)-axis.A reflection across the \(y\)-axis changes the sign of the \(x\)-values, so it is a horizontal reflection.Answer: C
2.
Item 6. Prepare for transformation the following equation: \(y=5(-4x+36)^3+2.5\).
Solution:Start with:\[y=5(-4x+36)^3+2.5\]Factor the coefficient of \(x\) from the expression inside the brackets:\[-4x+36=-4(x-9)\]Substitute back into the equation:\[y=5[-4(x-9)]^3+2.5\]Therefore, the prepared form is \(y=5[-4(x-9)]^3+2.5\).Answer: C
3.
Item 3. The graph of \(y=f(x)\) is shown. If the function is changed to \(y=f\left(\frac{1}{3}x\right)\), what will be point A's new coordinates?
Solution:Replacing \(x\) with \(\frac{1}{3}x\) causes a horizontal expansion by a factor of \(3\).This means every \(x\)-coordinate is multiplied by \(3\), while the \(y\)-coordinate stays the same.Point A is \((-1,6)\).\[(-1,6)\rightarrow(-1\cdot 3,6)=(-3,6)\]Therefore, the new coordinates are \((-3,6)\).Answer: C
4.
Item 2. Which of the following statement is false for an inverse reflection?
Solution:An inverse reflection reflects a graph across the line \(y=x\).Invariant points stay fixed, so they must lie on the line \(y=x\).For \((4,4)\), \(x=y\), so it is invariant.For \((-1,1)\), \(x eq y\), so it is not on the line \(y=x\).Therefore, the false statement is that \((-1,1)\) is an invariant point.Answer: C
5.
Item 5. If \((-6,1)\) is a point on the graph \(f(x)\), what point must be on the graph of \(y=2f(x)-3\)?
Solution:The transformation \(y=2f(x)-3\) changes only the \(y\)-value.The \(x\)-coordinate stays the same.Since \((-6,1)\) is on \(f(x)\), we use \(f(-6)=1\).\[y=2(1)-3=2-3=-1\]Therefore, the new point is \((-6,-1)\).Answer: C
6.
Item 4. The graph of \(y=f(x)\) is shown. If the function is changed to \(y=f\left(\frac{1}{2}x\right)\), then the transformed graph is best illustrated by:
Solution:Replacing \(x\) with \(\frac{1}{2}x\) causes a horizontal expansion by a factor of \(2\).Therefore, every \(x\)-coordinate is doubled, while every \(y\)-coordinate stays the same.For example, the point \((2,-1)\) becomes:\[(2,-1)\rightarrow(4,-1)\]The graph that shows this horizontal expansion is Graph B.Answer: B
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