Question

transformations

parent function: (y = \sqrt{x})

transformation: (y = a\sqrt{x - h} + k)

a: leading coefficient

+ : the graph ________ left to right

• : the graph ________ left to right

(reflection over the x-axis)

Explanation:

🆕 New Concept Discovered: Function Transformations
How changes to an equation shift, stretch, or flip its graph.

Step 1: Analyze the positive leading coefficient

The parent function is \( y = \sqrt{x} \).

When the leading coefficient \( a \) is positive (\( + \)), the graph starts at its endpoint and curves upward as it moves from left to right.

Therefore, the graph rises (or increases) from left to right.

Step 2: Analyze the negative leading coefficient

When the leading coefficient \( a \) is negative (\( - \)), the graph is reflected vertically over the \( x \)-axis.

Instead of curving upward, the graph starts at its endpoint and curves downward as it moves from left to right.

Therefore, the graph falls (or decreases) from left to right.

Answer:

• \( + \) : The graph rises (or increases) left to right
• \( - \) : The graph falls (or decreases) left to right