Question

sketch the graph of the equation.
$y = \sqrt{x + 4}$

choose the correct graph of the equation.

\\(\bigcirc\\) a.

\\(\bigcirc\\) b.

\\(\bigcirc\\) c.

\\(\bigcirc\\) d.

Explanation:

Step1: Analyze the domain

The function is \( y = \sqrt{x + 4} \). For the square root to be defined, the expression inside must be non - negative. So, \( x+4\geq0 \), which gives \( x\geq - 4 \).

Step2: Analyze the range

Since the square root of a non - negative number is non - negative, \( y=\sqrt{x + 4}\geq0 \).

Step3: Analyze the shape and key points

• When \( x=-4 \), \( y = \sqrt{-4 + 4}=0 \), so the graph passes through the point \((-4,0)\).
• When \( x = 0 \), \( y=\sqrt{0 + 4}=2 \), so the graph passes through the point \((0,2)\).
• The function \( y=\sqrt{x+4} \) is a square - root function. The parent function \( y = \sqrt{x}\) has a domain \( x\geq0 \) and range \( y\geq0 \), and it is a curve that starts at the origin \((0,0)\) and increases slowly. The function \( y=\sqrt{x + 4}\) is a horizontal shift of the parent function \( y=\sqrt{x}\) to the left by 4 units.
• Now, let's analyze the options:
• Option A: The graph seems to have a different shape (more like an exponential or a steeper curve) and does not match the square - root curve.
• Option B: This looks like a parabola (quadratic function) opening upwards, but our function is a square - root function, not a quadratic function.
• Option C: The graph starts at \( x=-4\) (since when \( x=-4,y = 0\)), and as \( x\) increases, \( y\) increases slowly, which is consistent with the square - root function \( y=\sqrt{x + 4}\).
• Option D: This also looks like a quadratic function (symmetric about the y - axis), which is not the case for our square - root function.

Answer:

C