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In the world of quadratic functions, a vertical stretch occurs when we scale a function upward or downward. Think of it like pulling or compressing a rubber band vertically. When you see a coefficient in front of the squared term, like the 5 in our function \[ g(x) = 5(x+3)^2 - 2 \], it indicates a vertical stretch. This coefficient affects how "tall" or "narrow" the parabola will be.

• When the coefficient is greater than 1, as in our case, the parabola undergoes a vertical stretch. It becomes narrower compared to the usual \( x^2 \).
• If it were between 0 and 1, it would cause a vertical compression, making the parabola wider.
• A negative coefficient would flip the parabola upside down, but still stretch or compress it based on the value.

Thus, in our example, because the coefficient is 5, the graph becomes five times taller than the graph of \( x^2 \). This means any point on the curve is five times the height of the corresponding point on the basic parabola.

A horizontal shift in a quadratic function can be identified by looking at the term inside the parenthesis. It tells us how far left or right to move the entire graph.In our equation \[ (x + 3) \], we see a "+3" inside the parentheses. This indicates a horizontal shift, but contrary to what you might initially think, it actually moves the parabola 3 units to the left, not the right.

• If there is a "+" inside \( (x + a) \), it shifts \( a \) units left.
• If there is a "-" inside \( (x - a) \), it shifts \( a \) units right.

The reasoning behind this is due to how functions transform input values. By shifting left 3 units in our function, \((x+3)^2\) effectively moves every point of the \( x^2 \) parabola 3 steps to the left along the x-axis.

Vertical shifts in quadratic functions are straightforward and happen outside the squared term. They move the graph up or down on the y-axis without altering the shape of the parabola.In our function \[ g(x) = 5(x+3)^2 - 2 \], the \( -2 \) at the end tells us that every point on the graph of the parabola is shifted 2 units down.

• Positive constants move the graph upwards.
• Negative constants move the graph downwards.

This particular transformation does not affect the wideness or narrowness of the parabola, only the vertical position. So, after accounting for the vertical stretch and horizontal shift, you pull the entire graph down by 2 units. This means that its vertex, which is originally at \((0, 0)\), gets relocated first by shifting 3 units left, then stretched, and finally 2 units downward to \((-3, -2)\).