# 008A Sample Final A, Question 3

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**Question:** a) Find the vertex, standard graphing form, and X-intercept for

b) Sketch the graph. Provide the focus and directrix.

Note: In this problem, what is referred to as standard graphing form is the vertex form, in case you search on the internet.

Foundations: |
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1) What type of graph is this? (line, parabola, circle, etc.) |

2) What is the process for transforming the function into the standard graphing form? |

3) After we have the standard graphing form how do you find the X-intercept, and vertex? |

4) Moving on to part b) How do we find a point on the graph? |

5) From the standard graphing form how do we obtain relevant information about the focus and directrix? |

Answers: |

1) The function is a parabola. Some of the hints: We are asked to find the vertex, and directrix. Also only one variable, of x and y, is squared. |

2) First we complete the square. Then we divide by the coefficient of x. |

3) To find the X-intercept, replace y with 0 and solve for x. Since the parabola is in standard graphing form, the vertex of is (h, k). |

4) To find a point, we can either use the symmetry of a parabola or plug in another value for x. |

5) From the equation , we use the equation to find p. P is both the distance from the vertex to the focus and the distance from the vertex to the directrix. |

Solution:

Step 1: |
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There are two ways to determine the standard graphing form. |

Regardless of the method the first step is the same: subtract 2 from both sides to yield |

Method 1: |

Divide both sides by -3 to make the coefficient of , 1. This means |

Complete the square to get |

Multiply both sides by -3 so , and simplify the left side to yield |

Method 2: |

Instead of dividing by -3 we factor it out of the right hand side to get . |

Now we complete the square inside the parenthesis and add -3 to the left hand side resulting in |

We mention here that some instructors/professors are particular about which of these two methods you use. |

Step 2: |
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Since the parabola is in standard graphing form we can read off the vertex, which is (5, -1). |

We get the X-intercept by replacing y with 0 and solving for x. So , and the X-intercept is (2, 0). |

Step 3: |
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Now we need the value of p using the relation , where a = -3. |

So , and . |

Step 4: |
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Since , the parabola opens left. Since the focus is inside the parabola, and p tells us the focus is from the vertex, the focus is at . |

We also know that the directrix is a vertical line on the outside of the parabola with the distance from the directrix to the vertex being p. Thus the directrix is |

Final Answer: |
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Vertex: (5, -1), standard graphing form: , X-intercept: (2, 0), focus: , directrix: |