Question

move at least one of the 3 guide points below to complete the graph of $y = \sqrt{x + 2} - 5$. moving the red points changes the vertical stretch or compression. moving the blue point shifts the function left/right/up/down. click the buttons below to start over or reflect over the x-axis. reset reflect over x-axis

Explanation:

Step1: Identify the parent function

The parent function is \( y = \sqrt{x} \), which has a vertex at \( (0, 0) \), and passes through \( (1, 1) \), \( (4, 2) \) etc.

Step2: Analyze the transformations

For the function \( y = \sqrt{x + 2}-5 \), the \( x + 2 \) inside the square root shifts the graph 2 units to the left (horizontal shift: \( h=-2 \) in \( y=\sqrt{x - h}+k \)), and the \( -5 \) shifts it 5 units down (vertical shift: \( k = - 5 \)).

Step3: Find key points of the transformed function

• Vertex: For \( y=\sqrt{x+2}-5 \), set \( x + 2=0\Rightarrow x=-2 \), \( y = - 5 \). So the vertex (blue point) should be at \( (-2,-5) \).
• Other points: Take \( x=-1 \): \( y=\sqrt{-1 + 2}-5=\sqrt{1}-5=1 - 5=-4 \). So one red point at \( (-1,-4) \). Take \( x = 2 \): \( y=\sqrt{2 + 2}-5=\sqrt{4}-5=2 - 5=-3 \). So another red point at \( (2,-3) \).

Step4: Move the points

• Move the blue point (vertex) from \( (0,0) \) to \( (-2,-5) \).
• Move the first red point (originally at \( (1,1) \)) to \( (-1,-4) \).
• Move the second red point (originally at \( (4,2) \)) to \( (2,-3) \).

Answer:

Move the blue point to \((-2, -5)\), one red point to \((-1, -4)\), and the other red point to \((2, -3)\) to graph \(y=\sqrt{x + 2}-5\).