Function g is a transformation of the parent function f(x) = x2. The graph of g is a translation left 4 units and down 2 units of the graph of f.
Write the equation for g in the form y = ax2 + bx + c.

The parent function is \( f(x) = x^2 \). The transformation described involves translating the graph of \( f(x) \) left by 4 units and down by 2 units.

### Step 1: Translation left 4 units
A leftward translation of 4 units changes the input \( x \) to \( x + 4 \). So, the function becomes:
\[
g(x) = (x + 4)^2
\]

### Step 2: Translation down 2 units
A downward translation of 2 units subtracts 2 from the function:
\[
g(x) = (x + 4)^2 - 2
\]

### Step 3: Expand and simplify the equation
Now, expand \( (x + 4)^2 \):
\[
(x + 4)^2 = x^2 + 8x + 16
\]
Thus, the function becomes:
\[
g(x) = x^2 + 8x + 16 - 2
\]
Simplify:
\[
g(x) = x^2 + 8x + 14
\]

### Final equation:
\[
g(x) = x^2 + 8x + 14
\]

This is the equation of the transformed function in the form \( y = ax^2 + bx + c \).

Do you want more details or have any questions?

Here are 5 related questions to expand on this concept:
1. What would be the equation if the graph were shifted right 3 units and up 5 units?
2. How does a vertical stretch or compression affect the graph's equation?
3. What is the vertex of the function \( g(x) = x^2 + 8x + 14 \)?
4. How would a reflection over the x-axis change the equation of the function?
5. How would you find the x-intercepts of \( g(x) \)?

**Tip**: Always expand and simplify the quadratic equation after performing translations to ensure it is in standard form.

Function g is a transformation of the parent function f(x) = x^2. The graph of g is a translation left 4 units and down 2 units of the graph of f. Write the equation for g in the form y = ax^2 + bx + c.

Learn how to find the equation of the transformed quadratic function by translating the graph of f(x) = x^2 left by 4 units and down by 2 units. Follow this step-by-step solution, perfect for Grades 9-11 students exploring function transformations and quadratic equations.

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