Question

question move at least one of the 5 guide points below to complete the graph of y = x² - 2. 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.

Explanation:

Step1: Analyze the parent function

The parent function is \( y = x^2 \), which is a parabola opening upwards with vertex at \((0,0)\). The target function is \( y = x^2 - 2 \), which is a vertical shift of the parent function.

Step2: Determine the transformation

For the function \( y = x^2 - 2 \), compared to \( y = x^2 \), we need to shift the graph of \( y = x^2 \) down by 2 units. This means moving the blue vertex point (which is at \((0,0)\) for \( y = x^2 \)) down to \((0, -2)\). Also, the red points (which are on \( y = x^2 \)) will also shift down by 2 units. For example, a red point at \((1,1)\) on \( y = x^2 \) will move to \((1, 1 - 2)=(1, -1)\), a red point at \((2,4)\) will move to \((2, 4 - 2)=(2, 2)\), etc.

Step3: Execute the transformation (conceptually)

To graph \( y = x^2 - 2 \), we shift the vertex (blue point) of \( y = x^2 \) (originally at \((0,0)\)) down 2 units to \((0, -2)\). Then, we adjust the red points accordingly by subtracting 2 from their \( y \)-coordinates (since the transformation is vertical shift down by 2).

Answer:

To graph \( y = x^2 - 2 \) from \( y = x^2 \):

1. Move the blue vertex point (originally at \((0,0)\)) down 2 units to \((0, -2)\).
2. Move each red point down 2 units (e.g., \((1,1)\) to \((1, -1)\), \((2,4)\) to \((2, 2)\), \((-1,1)\) to \((-1, -1)\), \((-2,4)\) to \((-2, 2)\)).

(Note: Since this is an interactive graphing problem, the key is to perform the vertical shift down by 2 units on the vertex and the other points.)