The graph below is the graph of the absolute value function \(\displaystyle f(x)= |x|\), which pairs each \(\displaystyle x\)-coordinate with its absolute value.

The given graph is the same as the above graph, except that each \(\displaystyle x\)-coordinate is paired with the \(\displaystyle y\)-coordinate three times that with which it is paired in the above graph. Therefore, the equation graphed is 

\(\displaystyle y = 3 \cdot |x|\)

or

\(\displaystyle y = |3x|\)

The graph of a function \(\displaystyle y = f(x)\) shifted right three units is the graph of \(\displaystyle y = f(x-3)\). In this graph, \(\displaystyle f(x)= x^{2}+ 2x - 17\), so the graph formed by the transformation is

\(\displaystyle f(x-3)= \left (x-3 \right ) ^{2}+ 2 (x-3) - 17\)

\(\displaystyle f(x-3)= x^{2}-6x+9+ 2x-6 - 17\)

\(\displaystyle f(x-3)= x^{2}-6x+ 2x+9-6 - 17\)

\(\displaystyle f(x-3)= x^{2}-4x -14\)

The correct equation is \(\displaystyle y= x^{2}-4x -14\).

The reflection of the graph of the equation \(\displaystyle y = f(x)\) about the origin is the graph of the equation \(\displaystyle -y = f(-x)\), so replace \(\displaystyle x\) with \(\displaystyle -x\) and \(\displaystyle y\) with \(\displaystyle -y\):

\(\displaystyle y = 2x^2 + 5x - 17\)

becomes

\(\displaystyle - y = 2 (-x) ^2 + 5 (-x) - 17\)

\(\displaystyle - y = 2 x^2 -5x - 17\)

\(\displaystyle -(-y) = - (2 x^2 -5x - 17)\)

\(\displaystyle y= - 2 x^2 +5x +17\)

The reflection of the graph of the equation \(\displaystyle y = f(x)\) about the \(\displaystyle x\)-axis is the graph of the equation \(\displaystyle -y = f(x)\), so replace \(\displaystyle y\) with \(\displaystyle -y\):

\(\displaystyle y = 2x^2 + 5x - 17\)

becomes

\(\displaystyle -y = 2x^2 + 5x - 17\)

\(\displaystyle -(-y) = - (2x^{2 }+ 5x - 17)\)

\(\displaystyle y =- 2x^2 -5x +17\)

The reflection of the graph of the equation \(\displaystyle y = f(x)\) about the \(\displaystyle y\)-axis is the graph of the equation \(\displaystyle y = f(-x)\), so replace \(\displaystyle x\) with \(\displaystyle -x\):

\(\displaystyle y = 2x^2 + 5x - 17\)

becomes

\(\displaystyle y = 2 (-x) ^2 + 5 (-x) - 17\)

\(\displaystyle y = 2 x^2 -5x - 17\)

The reflection of the graph of the equation \(\displaystyle y = f(x)\) about the origin is the graph of the equation \(\displaystyle -y = f(-x)\), so replace \(\displaystyle x\) with \(\displaystyle -x\) and \(\displaystyle y\) with \(\displaystyle -y\):

\(\displaystyle y = |x+3| - 5\)

becomes

\(\displaystyle -y = |-x+3| - 5\)

\(\displaystyle -(-y) =-( |-x+3| - 5)\)

\(\displaystyle y=- |-x+3| +5\)

The absolute values of two expressions that are each other's opposites are equal, so \(\displaystyle |-x+3|\) is equivalent to \(\displaystyle |x-3|\). The expression can be rewritten as 

\(\displaystyle y=- |x-3| +5\).

The graph below is the graph of the absolute value function \(\displaystyle f(x)= |x|\), which pairs each \(\displaystyle x\)-coordinate with its absolute value.

The given graph is the above graph shifted right five units and upward three units. The graph of any function \(\displaystyle y = f(x)\) shifted right five units and upward three units is \(\displaystyle y = f(x+5)+3\), so the correct response is \(\displaystyle y = |x+5| + 3\).

The graph of a function \(\displaystyle y = f(x)\) shifted downward seven units is the graph of \(\displaystyle y = f(x) -7\). In this graph, \(\displaystyle f(x)= x^{2}+ 3x - 5\), so the graph formed by the transformation is that of the equation

\(\displaystyle y = ( x^{2}+ 3x - 5)-7\)

\(\displaystyle y = x^{2}+ 3x -12\)

The graph below is the graph of the absolute value function \(\displaystyle f(x)= |x|\), which pairs each \(\displaystyle x\)-coordinate with its absolute value.

The given graph is the above graph shifted right two units. The graph of \(\displaystyle y = f(x)\) shifted right two units is \(\displaystyle y = f(x-2)\), so the correct response is \(\displaystyle y = |x-2|\).

The greatest integer function, or floor function, \(\displaystyle y = \left \lfloor x\right \rfloor\), pairs each value of \(\displaystyle x\) with the greatest integer less than or equal to \(\displaystyle x\). Its graph is below.

The given graph is the above graph shifted downward four units. The graph of any function \(\displaystyle y = f(x)\) shifted downward four units is \(\displaystyle y = f(x) - 4\), so the given graph corresponds to equation \(\displaystyle y = \left \lfloor x\right \rfloor - 4\).