Write the steps to obtain the graph of the function y = 3(x − 1)2 + 5 from the graph y = x2

#### Solution

Step 1:
Draw the graph y = x2

 x 0 1 − 1 2 − 2 y 0 1 1 4 4

Step 2:
The graph of y = (x – 1)2 shifts to the right for one unit.

 x 0 1 − 1 2 − 2 3 y 1 0 4 1 9 4

The graph of y = (x – 1)2 shifts the graph

y = x2 to the right by 1 unit.

The graph of y = f(x – c), c > 0 causes the graph y = f(x) a shift to the right by c units.

Step 3:
The graph of y = 3(x – 1)2 compresses towards y-axis that is moves away from the x-axis since the multiplying factor is which is greater than 1.

 x 0 1 – 1 2 – 2 3 y 3 0 12 3 24 12

The graph of y = 3(x – 1)2 compresses the graph y = (x – 1)2 towards the y-axis that is moving away from the x-axis since the multiplying factor is greater than 1.

For the graph y = kf(x), If k is a positive constant greater than one, the graph moves away from the x-axis.

If k is a positive constant less than one, the graph moves towards the x-axis.

Step 4:
The graph of y = 3(x – 1)2 + 5 causes the shift to the upward for 5 units.

 x 0 1 – 1 2 – 2 3 y 8 5 17 8 32 17

The graph of y = 3(x – 1)2 + 5 causes the graph y = 3(x – 1)2 shifts to the upward for 5 units.

The graph of y = f(x) + d, d > 0 causes the graph y = f(x) a shift to the upward by d units.

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