Question

move at least one of the 5 guide points below to complete the graph of $y = x^2 - 5$. 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 and transformation

The parent function is \( y = x^2 \), which has its vertex at \( (0,0) \). The given function is \( y = x^2 - 5 \), which is a vertical shift down by 5 units. So the vertex (blue point) should be moved to \( (0, -5) \).

Step2: Check the red points (for vertical stretch/compression)

The coefficient of \( x^2 \) is 1, so there's no vertical stretch or compression. The red points on \( y = x^2 \) are at \( (1,1) \), \( (-1,1) \), \( (2,4) \), \( (-2,4) \). For \( y = x^2 - 5 \), these points should be \( (1, 1 - 5) = (1, -4) \), \( (-1, -4) \), \( (2, 4 - 5) = (2, -1) \), \( (-2, -1) \). But the main shift is the vertex. So move the blue point (vertex) from \( (0,0) \) down to \( (0, -5) \).

Answer:

Move the blue vertex point to \( (0, -5) \) (and adjust red points accordingly: e.g., red points at \( x = 1, -1 \) move to \( y = -4 \), \( x = 2, -2 \) move to \( y = -1 \) if needed, but the key is shifting the vertex down 5 units).