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Which is the graph of y = 2 – square root of x?

On a coordinate plane, a curved line approaches the grid line at (3, 2.3), crosses the y-axis at (0, 1.5) and ends at (negative 2, 0).

On a coordinate plane, a curved line approaches the grid line at (negative 3, 2.3), crosses the y-axis at (0, 1.5) and ends at (2, 0).

On a coordinate plane, a curved line approaches the grid line at (3, 0.3), and ends at (0, 2).

On a coordinate plane, a curved line approaches the grid line at (3, negative 0.3) and ends at (0, negative 2).

1 answer

To analyze the graph of the function \( y = 2 - \sqrt{x} \), we can start by identifying some key characteristics:

  1. Domain: The square root function \( \sqrt{x} \) is only defined for \( x \geq 0 \). Therefore, the domain of the function \( y = 2 - \sqrt{x} \) is \( x \geq 0 \).

  2. Intercepts:

    • Y-intercept: When \( x = 0 \), \[ y = 2 - \sqrt{0} = 2 - 0 = 2. \] So, the graph crosses the y-axis at (0, 2).
    • X-intercept: Set \( y = 0 \): \[ 0 = 2 - \sqrt{x} \implies \sqrt{x} = 2 \implies x = 4. \] So, the graph crosses the x-axis at (4, 0).

  3. Behavior: As \( x \) increases, \( y \) will decrease since \( \sqrt{x} \) increases. The function will approach \( y = 2 \) as \( x \) approaches 0, and it will decrease without bound as \( x \) increases beyond 4 (approaching negative values as \( x \) becomes larger).

Given these points, we look for a graph that:

  • Crosses the y-axis at (0, 2).
  • Crosses the x-axis at (4, 0).
  • Exists only for \( x \geq 0 \).

None of the descriptions you provided approximate this behavior correctly, particularly none of them state that the graph crosses the y-axis at (0, 2) and the x-axis at (4, 0) while only existing in the first quadrant. Therefore, none of the options accurately represent the graph of \( y = 2 - \sqrt{x} \).