Graph the following function using transformations.
y equals 4 left parenthesis x minus 1 right parenthesis squared minus 3

To graph the function \( y = 4(x - 1)^2 - 3 \) using transformations, we can break it down step by step. This function is a transformed version of the basic quadratic function \( y = x^2 \).

### Step 1: Start with the parent function
The parent function is:
\[ y = x^2 \]
This is a basic parabola that opens upward with its vertex at \( (0, 0) \).

### Step 2: Apply horizontal shift
The term \( (x - 1) \) indicates a **horizontal shift** of the graph to the **right by 1 unit**.
Now, the vertex moves from \( (0, 0) \) to \( (1, 0) \).

### Step 3: Apply vertical stretch
The coefficient 4 in front of \( (x - 1)^2 \) means there is a **vertical stretch** by a factor of 4. This makes the parabola narrower compared to the standard \( y = x^2 \).

### Step 4: Apply vertical shift
The term \( -3 \) at the end of the function represents a **vertical shift**. The graph is shifted **downward by 3 units**.
Now, the vertex moves from \( (1, 0) \) to \( (1, -3) \).

### Summary of Transformations:
- Horizontal shift: Right 1 unit.
- Vertical stretch: Stretched by a factor of 4.
- Vertical shift: Down 3 units.

The final graph is a parabola with the vertex at \( (1, -3) \), opening upwards, and narrower than the standard parabola due to the vertical stretch.

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Graph the following function using transformations: y = 4(x - 1)^2 - 3

Learn how to graph the quadratic function y = 4(x - 1)^2 - 3 step by step using transformations. This problem covers key algebraic concepts, including horizontal and vertical shifts, vertical stretch, and the basic quadratic function. Suitable for students in Grades 9-11.

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