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Which statement ,begin emphasis,best,end emphasis, describes the effect on the parabola with an equation of y equals x squared plus 4 x when it is changed to y equals x squared plus 6 x?

Answer options with 4 options
A.
The parabola is shifted to the left 2 units.

B.
The parabola is shifted to the right 2 units.

C.
The parabola is shifted up 5 units and to the right 1 unit.

D.
The parabola is shifted down 5 units and to the left 1 unit.

3 answers

To determine the effect on the parabola represented by the equations \( y = x^2 + 4x \) and \( y = x^2 + 6x \), we can rewrite each equation in vertex form.

  1. For \( y = x^2 + 4x \):

    We complete the square: \[ y = (x^2 + 4x) = (x^2 + 4x + 4) - 4 = (x + 2)^2 - 4 \] The vertex is at the point \((-2, -4)\).

  2. For \( y = x^2 + 6x \):

    Again, we complete the square: \[ y = (x^2 + 6x) = (x^2 + 6x + 9) - 9 = (x + 3)^2 - 9 \] The vertex is at the point \((-3, -9)\).

Now we compare the vertices of the two parabolas.

  • The first parabola has a vertex at \((-2, -4)\), and the second parabola has a vertex at \((-3, -9)\).

From \((-2, -4)\) to \((-3, -9)\):

  • The horizontal shift is from \(-2\) to \(-3\), which is a shift to the left by \(1\) unit.
  • The vertical shift is from \(-4\) to \(-9\), which is a shift downward by \(5\) units.

Thus, the correct statement is: D. The parabola is shifted down 5 units and to the left 1 unit.

u sure

Let's go through the analysis step by step again to ensure clarity and correctness.

  1. Original Equation: \[ y = x^2 + 4x \]

    Completing the square: \[ y = (x^2 + 4x) = (x^2 + 4x + 4) - 4 = (x + 2)^2 - 4 \] Therefore, the vertex of this parabola is at \((-2, -4)\).

  2. Modified Equation: \[ y = x^2 + 6x \]

    Completing the square: \[ y = (x^2 + 6x) = (x^2 + 6x + 9) - 9 = (x + 3)^2 - 9 \] Thus, the vertex of this parabola is at \((-3, -9)\).

Now, let's compare the two vertices:

  • The first vertex is at \((-2, -4)\).
  • The second vertex is at \((-3, -9)\).

Shifts:

  • Horizontal Shift: From \(-2\) to \(-3\) is a shift to the left by \(1\) unit.
  • Vertical Shift: From \(-4\) to \(-9\) is a shift down by \(5\) units.

So, the statements about the shifts are verified:

  • The parabola shifted down \(5\) units and to the left \(1\) unit.

Therefore, the answer is indeed: D. The parabola is shifted down 5 units and to the left 1 unit.

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