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Compare the graphs and determine which one could be the graph of a square root function.(1 point)

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A coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A straight line is plotted passing through 3 marked points in the first and third quadrants.
Image with alt text: A coordinate plane's axes range from negative 4 to 4, both by 1-unit increments. A straight line is plotted passing through 3 marked points in the first and third quadrants.

A coordinate plane's x-axis ranges from negative 3 to 3 and its y-axis ranges from negative 1 to 4, both by 1-unit increments. A concave up parabola is drawn passing through 3 marked points in the first and second quadrants.
Image with alt text: A coordinate plane's x-axis ranges from negative 3 to 3 and its y-axis ranges from negative 1 to 4, both by 1-unit increments. A concave up parabola is drawn passing through 3 marked points in the first and second quadrants.

A coordinate plane's axes range from negative 3 to 3, both by 1-unit increments. An S-shaped curve is plotted passing through 3 marked points.
Image with alt text: A coordinate plane's axes range from negative 3 to 3, both by 1-unit increments. An S-shaped curve is plotted passing through 3 marked points.

A coordinate plane's x-axis ranges from negative 2 to 6 and its y-axis ranges from negative 2 to 4, both by 1-unit increments. A curve is plotted in the first quadrant. It passes through 3 marked points.
Image with alt text: A coordinate plane's x-axis ranges from negative 2 to 6 and its y-axis ranges from negative 2 to 4, both by 1-unit increments. A curve is plotted in the first quadrant. It passes through 3 marked points.
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1 answer

To determine which graph could represent a square root function, we need to understand the characteristics of a square root function, which can generally be represented in the form \( f(x) = \sqrt{x} \).

Key characteristics of a square root function:

  1. Domain: The function is defined for \( x \geq 0 \). Hence, the graph will only exist in the first quadrant and along the x-axis for non-negative values of x.

  2. Range: The outputs (y-values) are also non-negative, so y will be \( y \geq 0 \).

  3. Shape: The graph starts at the origin (0,0) and increases at a decreasing rate, curving upwards gradually.

Now, reviewing the provided descriptions:

  1. First graph: A straight line in the first and third quadrants. This cannot be a square root function because it does not exhibit the characteristic shape and includes negative x values.

  2. Second graph: A concave up parabola in the first and second quadrants. This is not a square root function since parabolas are quadratic, not square root functions.

  3. Third graph: An S-shaped curve. This sounds more like it could represent a polynomial or sine/cosine function but doesn’t specifically indicate properties of a square root function.

  4. Fourth graph: A curve in the first quadrant that passes through 3 marked points. If it starts from the origin and curves upward while remaining in the first quadrant, it has the potential to be the graph of a square root function.

Based on the characteristics of square root functions, the fourth graph is the most likely to represent a square root function.