Fall 2019 Math 115 Quiz 7 Show All Work

Fall 2019 Math 115 Quiz 7 Show All Work

Put the equation into the standard form of a parabola and find the vertex, focus, and directrix. The standard form of a parabola can be found in Section 7.3 of the textbook. Answers should be exact values and not approximations.

4y 2 - 4y + x = 1

Put the equation into the standard form, and find the vertex, focus, and directrix. Answers should be exact values and not approximations.

Paper For Above instruction

The task is to convert the given quadratic equation into the standard form of a parabola, and then identify key characteristics such as the vertex, focus, and directrix, all in exact form. The initial equation provided is 4y² - 4y + x = 1. This equation involves both quadratic and linear terms, making it necessary to complete the square for the y-terms and then rearrange into the vertex form of a parabola.

First, isolate x on one side to facilitate completing the square for the y-terms:

4y² - 4y = 1 - x

Factor out the 4 from the y terms:

4(y² - y) = 1 - x

Complete the square inside the parentheses. Take half of the coefficient of y, which is -1, resulting in -1/2, then square it, giving (1/2)² = 1/4. Add and subtract 1/4 inside the parentheses:

4(y² - y + 1/4 - 1/4) = 1 - x

Expressed as:

4[(y - 1/2)² - 1/4] = 1 - x

Distribute the 4:

4(y - 1/2)² - 4*(1/4) = 1 - x

which simplifies to:

4(y - 1/2)² - 1 = 1 - x

Add 1 to both sides:

4(y - 1/2)² = 2 - x

Rearranged for x:

x = 2 - 4(y - 1/2)²

This is the standard form of a parabola opening leftward (since x is expressed as the negative quadratic in y). The vertex form of a parabola opening horizontally is x = a(y - k)² + h, with vertex (h, k).

Comparing, we identify:

  • Vertex: (h, k) = (2, 1/2)
  • Standard form: x = -4(y - 1/2)² + 2

The coefficient -4 indicates that 4p = 1/4, meaning p = -1/16 (since in horizontal parabolas, 4p is the coefficient's magnitude, and negative indicates the parabola opens to the left).

Focus is located at (h + p, k):

Focus: (2 + (-1/16), 1/2) = (2 - 1/16, 1/2) = (31/16, 1/2)

The directrix is a vertical line located at x = h - p:

Directrix: x = 2 - (-1/16) = 2 + 1/16 = (32/16 + 1/16) = 33/16

In summary:

  • Standard form: x = -4(y - 1/2)² + 2
  • Vertex: (2, 1/2)
  • Focus: (31/16, 1/2)
  • Directrix: x = 33/16

These results are exact and fully describe the parabola’s position and shape as derived from the original equation.

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