Question

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how does parabola b, ( y = 5(x + 2)^2 ), compare to parabola a, the quadratic parent function?
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options:

• parabola b is narrower than parabola a and shifted to the left of the origin
• parabola b is narrower and its vertex is 2 units to the right of the origin
• parabola b is wider and the vertex is 5 units above the x - axis
• parabola b is wider than parabola a and shifted down

Explanation:

Step1: Recall the parent parabola

The parent quadratic function (parabola A) is \( y = x^2 \), with vertex at \((0,0)\) and a vertical stretch factor of \(1\).

Step2: Analyze the given parabola B

The equation of parabola B is \( y = 5(x + 2)^2 \).

• Vertical Stretch: The coefficient \(5\) (greater than \(1\)) means the parabola is vertically stretched, so it is narrower than the parent parabola (since a larger absolute coefficient makes the parabola narrower).
• Horizontal Shift: The \((x + 2)\) inside the square means the graph is shifted \(2\) units to the left (because the shift is opposite to the sign: \(x - h\) shifts right \(h\), so \(x + 2=x-(-2)\) shifts left \(2\) units). The vertex of \(y = 5(x + 2)^2\) is \((-2,0)\), which is \(2\) units to the left of the origin \((0,0)\).

Now, let's check the options:

• Option 1: "Parabola B is narrower than parabola A and shifted to the left of the origin" matches our analysis (narrower due to \(|5|>1\), shifted left \(2\) units).
• Option 2: The vertex is at \((-2,0)\), not \(2\) units right, and it's narrower (not wider), so this is wrong.
• Option 3: The vertex is at \((-2,0)\) (on the x - axis, \(y = 0\)), not \(5\) units above, and it's narrower, so wrong.
• Option 4: The parabola is narrower (not wider) and shifted left (not down), so wrong.

Answer:

Parabola B is narrower than parabola A and shifted to the left of the origin (the first option among the given choices).