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In the world of graph transformations, a horizontal shift moves the entire graph left or right along the x-axis. This occurs when we adjust the x-value inside a function. In our problem, we're starting with the function \(y = x^2\). A transformation of this function is called a horizontal shift when it changes to \(y = (x + 3)^2\). The "+3" inside the square suggests that each point on the graph will move 3 units to the left.

This is counterintuitive at first, since a positive number typically indicates moving forward or to the right. However, adding inside the function, as in \((x + 3)\), indicates a shift to the left because it affects the input to the function, moving the curve in the opposite direction of the sign. Remember:

• If you have \((x + a)\), the graph shifts left by \(a\) units.
• If you have \((x - a)\), the graph shifts right by \(a\) units.

This means for our transformation, we start by shifting the graph of \(y = x^2\) 3 units to the left to get \(y = (x + 3)^2\).

Vertical scaling changes the 'height' of the graph, making it stretch or shrink vertically. This adjustment is applied by multiplying the function by a constant factor. In our example, we transform \(y = (x + 3)^2\) to \(y = 4(x + 3)^2\) by multiplying every y-value by 4. Let's break it down:

Imagine each point on the graph of \((x + 3)^2\) is stretched away from the x-axis. The y-coordinates are quadrupled (since 4 is our scaling factor), so the graph appears taller. If this factor had been a fraction, such as 0.5, the graph would shrink instead, as all y-values would be halved.

To summarize, vertical scaling involves:

• Stretching the graph vertically if multiplied by a number greater than 1.
• Shrinking the graph vertically if multiplied by a number between 0 and 1.

In our transformation, we take the graph from \(y = (x + 3)^2\) and stretch it vertically by a factor of four, resulting in the graph of \(y = 4(x + 3)^2\).

A vertical shift involves moving the graph up or down along the y-axis. This occurs when we add or subtract a value from the entire function. For our problem, the transformation \(y = 4(x + 3)^2\) to \(y = 4(x + 3)^2 + 6\) involves a vertical shift. The "+6" indicates we move the entire graph upward by 6 units.

This is a straightforward modification: adding a constant to the entire function shifts the graph upward, while subtracting shifts it downward. In this case, every point on the graph of \(y = 4(x + 3)^2\) goes up 6 units. To remember:

• Add "+c" to move up by \(c\) units.
• Add "-c" to move down by \(c\) units.

Thus, our final step in the transformation is moving the graph of \(y = 4(x + 3)^2\) upward by 6 units, completing the sequence of transformations to obtain \(y = 4(x + 3)^2 + 6\).