How do stretches and reflections change a function graph?

Check where the multiplier is placed. In g(x)=a f(x), it multiplies the output, so it changes the graph vertically: each point (x,y) on f becomes (x,ay) on g. The x-coordinate stays put. If a is greater than 1, points move farther from the x-axis and the graph is stretched vertically. If 0<a<1, they move toward the x-axis and the graph is compressed vertically. [S1]

Use the parent function f(x)=√x. It has points (0,0), (1,1), and (4,2). For g(x)=2√x, those points become (0,0), (1,2), and (4,4). The graph is taller, but its x-values have not changed.

For g(x)=a f(x), look at both the size and sign of a. A negative multiplier reverses the sign of every output, reflecting the graph across the x-axis. Its absolute value determines any additional vertical scaling. For example, g(x)=−2√x sends (1,1) to (1,−2) and (4,2) to (4,−4): the points are below the x-axis and twice as far from it as before. [S1]

A useful coordinate check is: original point (x,y) becomes (x,ay). A point on the x-axis, where y=0, stays there. This example shows the rule for the square-root function; it does not mean every graph has the same shape or domain.

No. In f(2x), the 2 is inside the input, so it changes horizontal positions, not output heights. To match an original point (u,f(u)), solve 2x=u: the corresponding point is (u/2,f(u)). For √x, the original point (4,2) therefore appears at (2,2) on √(2x). The graph is compressed horizontally toward the y-axis. In general, a positive input factor greater than 1 compresses horizontally; a negative input factor also reflects across the y-axis. [S1]

Compare the two formulas at that same original point: 2√x moves (4,2) to (4,4), changing height; √(2x) moves it to (2,2), changing horizontal position. This point-tracking method helps avoid relying on whether a curve merely looks wider or taller.

First identify whether the number multiplies the whole function or is inside its input. Then track one or two known points before sketching. For example, starting with f(x)=√x and point (1,1):

• 3f(x): (1,1) becomes (1,3), a vertical stretch.
• −f(x): (1,1) becomes (1,−1), an x-axis reflection.
• f(3x): (1,1) corresponds to (1/3,1), a horizontal compression.
• f(−x): the input reversal reflects across the y-axis; for √x, the real-valued domain changes, so do not assume the whole original curve simply appears on both sides.

The general transformation rules describe how coordinates relate where the functions are defined. A quick sketch should also respect the original function's domain. [S1]