Scarleth27viz
Answered

Write the following in their general form and identify the vertex of the graph

1. f(x)=(x+3) ²+6
2. f(x) = -2(x+5)²+9
3. f(x)=-(x-7) ²- 2​

Sagot :

KAntoineDoixviz

✒️ QUADRATIC FUNC.

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[tex] \large\underline{\mathbb{DIRECTIONS}:} [/tex]

» Write the following in their general form and identify the vertex of the graph.

  • 1. f(x) = (x+3)² + 6
  • 2. f(x) = -2(x+5)² + 9
  • 3. f(x)= -(x-7)² - 2

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[tex] \large\underline{\mathbb{ANSWER}:} [/tex]

General Form:

[tex] \qquad \large \: \rm 1) \; f(x) = x^2 + 6x + 15 [/tex]

[tex] \qquad \large \: \rm 2) \; f(x) = \text-2x^2 - 20x - 41 [/tex]

[tex] \qquad \large \: \rm 3) \; f(x) = \text-x^2 + 14x - 51 [/tex]

Vertices:

[tex] \qquad \large \: \rm 1) \; (\text-3,6) [/tex]

[tex] \qquad \large \: \rm 2) \; (\text-5,9) [/tex]

[tex] \qquad \large \: \rm 3) \; (7,\text-2) [/tex]

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[tex] \large\underline{\mathbb{SOLUTION}:} [/tex]

» Since the quadratic function is already in vertex form, we can find its vertex as (h, k).

  • [tex] f(x) = a(x - h)^2 + k [/tex]

» After taking the values of h and k, we can now rearrange the function in general form.

  • [tex] f(x) = ax^2 + bx + c [/tex]

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Number 1:

» Since h is -3 and k is 6, then the vertex of the parabola is at (-3, 6). Now rearrange it in general form.

  • [tex] f(x) = (x + 3)^2 + 6 [/tex]

  • [tex] f(x) = x^2 + 6x + 9 + 6 [/tex]

  • [tex] f(x) = x^2 + 6x + 15 [/tex]

[tex] \therefore [/tex] f(x) = x² + 6x + 15 is the general form of the given quadratic function.

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Number 2:

» Since h is -5 and k is 9, then the vertex of the parabola is at (-5, 9). Now rearrange it in general form.

  • [tex] f(x) = \text-2(x + 5)^2 + 9 [/tex]

  • [tex] f(x) = \text-2(x^2 + 10x + 25) + 9 [/tex]

  • [tex] f(x) = \text-2x^2 - 20x - 50 + 9 [/tex]

  • [tex] f(x) = \text-2x^2 - 20x - 41 [/tex]

[tex] \therefore [/tex] f(x) = -2x² - 20x - 41 is the general form of the given quadratic function.

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Number 3:

» Since h is 7 and k is -2, then the vertex of the parabola is at (7, -2). Now rearrange it in general form.

  • [tex] f(x) = \text-(x - 7)^2 - 2 [/tex]

  • [tex] f(x) = \text-(x^2 - 14x + 49) - 2 [/tex]

  • [tex] f(x) = \text-x^2 + 14x - 49 - 2 [/tex]

  • [tex] f(x) = \text-x^2 + 14x - 51 [/tex]

[tex] \therefore [/tex] f(x) = -x² + 14x - 51 is the general form of the given quadratic function.

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