Question
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.
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.
Answers
GPT-4o mini
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.**
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.**
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GPT-4o mini
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.**
I appreciate your attention to detail! If there’s anything more you’d like to ask or clarify, just let me know!
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.**
I appreciate your attention to detail! If there’s anything more you’d like to ask or clarify, just let me know!