Given the graph of a function with multiple translations write the new function rule for the translated graph

Question

The graphs below shows g(x) which translation of f(x) =x^2. Write the function rule for g(x). Write your answer the form of a(x-h)^2+k Remember to use another point solve for a
The graph is an upward u shape with points (-1,-2), (-4,1),(2,1)

Answer 1

To write the new function rule for the translated graph \( g(x) \) in the form \( a(x-h)^2 + k \), we first need to determine the values of \( h \) and \( k \) using the vertex form of a quadratic function.

Step 1: Determine the Vertex

The vertex of the graph represents the minimum point for a parabola that opens upwards. Based on the points provided:

• The points provided are: (-1, -2), (-4, 1), (2, 1).
• The point (-1, -2) appears to have the lowest y-coordinate, suggesting that it might be the vertex.

This means:

• \( h = -1 \)
• \( k = -2 \)

Step 2: Write Initial Form

Now, we can express the function in the form \( g(x) = a(x - h)^2 + k \): \[ g(x) = a(x + 1)^2 - 2 \]

Step 3: Solve for \( a \) Using Another Point

Next, we will use another point from the graph to find the value of \( a \). Let's use the point (2, 1):

Substituting \( (x, y) = (2, 1) \) into the function: \[ 1 = a(2 + 1)^2 - 2 \]

\[ 1 = a(3)^2 - 2 \]

\[ 1 = 9a - 2 \]

Add 2 to both sides: \[ 1 + 2 = 9a \]

\[ 3 = 9a \]

Dividing both sides by 9: \[ a = \frac{1}{3} \]

Step 4: Write the Final Function Rule

Now we can write the final function rule for \( g(x) \): \[ g(x) = \frac{1}{3}(x + 1)^2 - 2 \]

This is the function rule for the translated graph \( g(x) \).