Question

the graph of ( y = x ) is translated down.
the graph of ( y = x ) is reflected about the ( y )-axis.
the graph of ( y = x ) is vertically stretched.
the graph of ( y = x ) is translated up.
the graph of ( y = x ) is vertically compressed.
the graph of ( y = x ) is three times the steepness.
( y = x + 1 )
( y = x - \frac{1}{2} )
( y = x )
( y = x - 2 )
( y = x + 5 )
( y = \frac{1}{2}x )
( y = x + 8 )
( y = 3x )
( h(x) = -x )
( h(x) = x - 3 )
( g(x) = x + 7 )
( g(x) = 10x )
( g(x) = \frac{1}{3}x )
( g(x) = 0.5x )
( y = \frac{1}{3}x )
( g(x) = x - 1.2 )
( g(x) = \frac{2}{3}x )
( g(x) = 2x )
( y = \frac{1}{4}x )
( y = \frac{1}{4}x )

Explanation:

Step1: Match vertical compression

A vertical compression of $y=x$ uses $g(x)=kx$ where $0The graph of $y=x$ is vertically compressed.

Step2: Match vertical stretch

A vertical stretch of $y=x$ uses $g(x)=kx$ where $k>1$. $g(x)=10x$ fits, so pair with The graph of $y=x$ is vertically stretched.

Step3: Match 3x steepness

A graph 3 times as steep as $y=x$ has slope 3. $y=3x$ fits, so pair with The graph of $y=x$ is three times as steep.

Step4: Match reflection over y-axis

Reflecting $y=x$ over the y-axis gives $y=-x$. $y=-x$ fits, so pair with The graph of $y=x$ is reflected about the y-axis.

Step5: Match translation down

Translating $y=x$ down uses $y=x-c$ ($c>0$). $g(x)=x-1.2$ fits, so pair with The graph of $y=x$ is translated down.

Step6: Match translation up

Translating $y=x$ up uses $y=x+c$ ($c>0$). $y=x+7$ fits, so pair with The graph of $y=x$ is translated up.

Step7: Remaining pairings

• $y=x+1$ ↔ $g(x)=x+5$ (both vertical up translations)
• $y=\frac{1}{3}x$ ↔ $y=\frac{1}{4}x$ (both vertical compressions)
• $y=x-\frac{1}{2}$ ↔ $y=x-2$ (both vertical down translations)
• $y=x+8$ ↔ $g(x)=x-3$ (both vertical translations)
• $g(x)=\frac{2}{3}x$ ↔ $y=\frac{5}{4}x$ (both vertical stretches/compressions)
• $g(x)=-x$ ↔ $y=-x$ (both reflections over y-axis)
• $g(x)=2x$ ↔ $g(x)=\frac{5}{7}x$ (both vertical stretches/compressions)

Answer:

1. $y=x+1$ ↔ $g(x)=x+5$
2. $y=x-\frac{1}{2}$ ↔ $y=x-2$
3. $y=-x$ ↔ $g(x)=-x$
4. $y=\frac{1}{3}x$ ↔ $y=\frac{1}{4}x$
5. $y=x+8$ ↔ $g(x)=x-3$
6. $y=3x$ ↔ The graph of $y=x$ is three times as steep.
7. $g(x)=10x$ ↔ The graph of $y=x$ is vertically stretched.
8. $g(x)=\frac{2}{3}x$ ↔ $y=\frac{5}{4}x$
9. $g(x)=x+7$ ↔ The graph of $y=x$ is translated up.
10. $g(x)=0.8x$ ↔ The graph of $y=x$ is vertically compressed.
11. $g(x)=\frac{5}{7}x$ ↔ $g(x)=2x$
12. $g(x)=x-1.2$ ↔ The graph of $y=x$ is translated down.
13. $y=-x$ ↔ The graph of $y=x$ is reflected about the y-axis.
14. $y=x+5$ ↔ $y=x+1$
15. $g(x)=x-3$ ↔ $y=x+8$
16. $y=\frac{1}{4}x$ ↔ $y=\frac{1}{3}x$
17. $y=\frac{5}{4}x$ ↔ $g(x)=\frac{2}{3}x$