Question

graph the function below by dragging the points.
$f(x) = \sqrt{x + 5} - 1$
show your work here

Explanation:

Step1: Identify the parent function

The parent function is \( y = \sqrt{x} \), which has a domain \( x \geq 0 \) and range \( y \geq 0 \), with a starting point at \( (0, 0) \).

Step2: Analyze the horizontal shift

For the function \( f(x) = \sqrt{x + 5} - 1 \), the \( x + 5 \) inside the square root indicates a horizontal shift. The formula for horizontal shift is \( \sqrt{x - h} \), so comparing \( \sqrt{x + 5} = \sqrt{x - (-5)} \), we see there is a shift of 5 units to the left. This changes the domain to \( x + 5 \geq 0 \) or \( x \geq -5 \).

Step3: Analyze the vertical shift

The \( -1 \) outside the square root indicates a vertical shift of 1 unit down. This changes the range to \( y \geq 0 - 1 \) or \( y \geq -1 \).

Step4: Find key points

• Starting point: For the parent function \( y = \sqrt{x} \), the starting point is \( (0, 0) \). After shifting 5 units left and 1 unit down, the new starting point is \( (-5, -1) \).
• Other points: Let's find a few more points.
• When \( x = -5 \), \( f(-5) = \sqrt{-5 + 5} - 1 = 0 - 1 = -1 \), so the point is \( (-5, -1) \).
• When \( x = -4 \), \( f(-4) = \sqrt{-4 + 5} - 1 = \sqrt{1} - 1 = 1 - 1 = 0 \), so the point is \( (-4, 0) \).
• When \( x = 4 \), \( f(4) = \sqrt{4 + 5} - 1 = \sqrt{9} - 1 = 3 - 1 = 2 \), so the point is \( (4, 2) \).

Step5: Plot the points

Plot the starting point \( (-5, -1) \), then use the shape of the square root function (increasing, concave down) to plot the other points like \( (-4, 0) \) and \( (4, 2) \), and draw the curve through these points, ensuring it follows the domain \( x \geq -5 \) and range \( y \geq -1 \).

Answer:

To graph \( f(x) = \sqrt{x + 5} - 1 \):

1. Start at the point \( (-5, -1) \) (due to horizontal shift left 5, vertical shift down 1 from the parent \( \sqrt{x} \)'s \( (0,0) \)).
2. Plot additional points (e.g., \( (-4, 0) \), \( (4, 2) \)) using the function.
3. Draw a smooth curve through these points, increasing and concave down, with domain \( x \geq -5 \) and range \( y \geq -1 \).