Question

suppose the graph of ( f ) is given. describe how the graph of each function can be obtained from the graph of ( f ).
(a) ( y=-f(x)+4 )
reflect about the ( y )-axis, then shift down 4 units
reflect about the ( y )-axis, then shift left 4 units
reflect about the ( x )-axis, then shift down 4 units
reflect about the ( x )-axis, then shift up 4 units
reflect about the ( y )-axis, then shift up 4 units
(b) ( y = 5 f(x)-4 )
stretch vertically by a factor of 5, then shift down 4 units
stretch vertically by a factor of 5, then shift up 4 units
shift down 4 units, then stretch vertically by a factor of 5
shrink vertically by a factor of 5, then shift down 4 units
stretch vertically by a factor of 4, then shift down 5 units

Explanation:

Part (a)

• For the transformation \(y = -f(x)+4\):
• The negative sign in front of \(f(x)\) (i.e., \(y=-f(x)\)) reflects the graph of \(y = f(x)\) about the \(x\) - axis. The rule for reflecting \(y = f(x)\) about the \(x\) - axis is \((x,y)\to(x, - y)\).
• The \(+4\) (i.e., \(y=-f(x)+4\)) shifts the graph of \(y=-f(x)\) vertically. The rule for vertical shift is: if \(y = g(x)+k\), when \(k>0\), the graph of \(y = g(x)\) is shifted up by \(k\) units. Here \(g(x)=-f(x)\) and \(k = 4\), so it shifts up 4 units.

Part (b)

• For the transformation \(y=5f(x)-4\):
• The coefficient \(5\) in front of \(f(x)\) (i.e., \(y = 5f(x)\)) causes a vertical stretch. The rule for vertical stretch is: if \(y=af(x)\) with \(|a|>1\), the graph of \(y = f(x)\) is stretched vertically by a factor of \(|a|\). Here \(a = 5\), so it is stretched vertically by a factor of 5.
• The \(-4\) (i.e., \(y=5f(x)-4\)) shifts the graph of \(y = 5f(x)\) vertically. Using the rule \(y = g(x)+k\) (where \(g(x)=5f(x)\) and \(k=-4\)), the graph is shifted down 4 units since \(k<0\).

Answer:

(a) reflect about the \(x\) - axis, then shift up 4 units
(b) stretch vertically by a factor of 5, then shift down 4 units