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.