How to find transformation for an analytic geometry equation - SSAT Upper Level Quantitative

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Question

What equation is graphed in the above figure?

Answer

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

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

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Question

What equation is graphed in the above figure?

Answer

The least integer function, or celing function, $f(x) = \left \lceil x\right \rceil$ , pairs each value of with the least integer greater than or equal to . Its graph is below.

The given graph is the above graph shifted upward four units. The graph of any function shifted upward four units is , so the given graph corresponds to equation $y = \left \lceil x\right \rceil+ 4$ .

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Question

Give the equation graphed in the above figure.

Answer

The graph below is the graph of the absolute value function , which pairs each -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 shifted right five units and upward three units is , so the correct response is .

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Question

Give the equation graphed in the above figure.

Answer

The graph below is the graph of the absolute value function , which pairs each -coordinate with its absolute value.

The given graph is the above graph shifted right two units. The graph of shifted right two units is , so the correct response is .

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Question

Give the equation graphed in the above figure.

Answer

The graph below is the graph of the absolute value function , which pairs each -coordinate with its absolute value.

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

$y = 3 \cdot |x|$

or

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Question

If the graph of the equation is reflected about the -axis, what will the equation of the resulting graph be?

Answer

The reflection of the graph of the equation about the -axis is the graph of the equation , so replace with :

becomes

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

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Question

If the graph of the equation $y= x^{2}+ 2x - 17$ is shifted right three units on the coordinate plane, what will be the equation of the resulting graph?

Answer

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

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

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

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

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

The correct equation is $y= x^{2}-4x -14$ .

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Question

If the graph of the equation $y= x^{2}+ 3x - 5$ is shifted downward seven units on the coordinate plane, what will be the equation of the resulting graph?

Answer

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

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

$y = x^{2}+ 3x -12$

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Question

If the graph of the equation is reflected about the -axis, what will the equation of the resulting graph be?

Answer

The reflection of the graph of the equation about the -axis is the graph of the equation , so replace with :

becomes

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Question

If the graph of the equation is reflected about the origin, what will the equation of the resulting graph be?

Answer

The reflection of the graph of the equation about the origin is the graph of the equation , so replace with and with :

becomes

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Question

Give the equation graphed in the above figure.

Answer

The graph below is the graph of the absolute value function , which pairs each -coordinate with its absolute value.

The given graph is the above graph shifted upward five units. The graph of any function shifted upward five units is , so the correct response is .

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Question

If the graph of the equation is reflected about the origin, what will the equation of the resulting graph be?

Answer

The reflection of the graph of the equation about the origin is the graph of the equation , so replace with and with :

becomes

The absolute values of two expressions that are each other's opposites are equal, so is equivalent to . The expression can be rewritten as

.

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Question

What equation is graphed in the above figure?

Answer

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

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

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Question

If the graph of the equation is reflected about the -axis, what will the equation of the resulting graph be?

Answer

The reflection of the graph of the equation about the -axis is the graph of the equation , so replace with :

becomes

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

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Question

What equation is graphed in the above figure?

Answer

The least integer function, or celing function, $f(x) = \left \lceil x\right \rceil$ , pairs each value of with the least integer greater than or equal to . Its graph is below.

The given graph is the above graph shifted upward four units. The graph of any function shifted upward four units is , so the given graph corresponds to equation $y = \left \lceil x\right \rceil+ 4$ .

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Question

Give the equation graphed in the above figure.

Answer

The graph below is the graph of the absolute value function , which pairs each -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 shifted right five units and upward three units is , so the correct response is .

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Question

Give the equation graphed in the above figure.

Answer

The graph below is the graph of the absolute value function , which pairs each -coordinate with its absolute value.

The given graph is the above graph shifted right two units. The graph of shifted right two units is , so the correct response is .

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Question

Give the equation graphed in the above figure.

Answer

The graph below is the graph of the absolute value function , which pairs each -coordinate with its absolute value.

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

$y = 3 \cdot |x|$

or

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Question

If the graph of the equation $y= x^{2}+ 2x - 17$ is shifted right three units on the coordinate plane, what will be the equation of the resulting graph?

Answer

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

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

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

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

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

The correct equation is $y= x^{2}-4x -14$ .

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Question

If the graph of the equation $y= x^{2}+ 3x - 5$ is shifted downward seven units on the coordinate plane, what will be the equation of the resulting graph?

Answer

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

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

$y = x^{2}+ 3x -12$

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