Section 3.5 solutions:
1) f(x) = (x-2)2 + 5 The (x-2)2 will shift the graph 2 units to the right. The +5 will shift the graph up 5
units.
Answer: shifted right 2 units and up 5 units
3) f(x) = √𝑥 + 3 – 5 : the √𝑥 + 3 will shift the graph left 3 units. The -5 will move the graph down 5
units.
Answer: shifted 3 units to the left and 5 units down.
5) 𝑓(𝑥) = |𝑥 − 3| + 1: The |𝑥 − 3| will shift the graph right 3 units. The +1 will shift the graph up 1
unit.
Answer: shifted right 3 units, up 1 unit
7) f(x) = -(x-3)2 + 2: The negative in front of the parenthesis will reflect over the x-axis. The -3 inside the
parenthesis will shift right 3 units and the +2 will move it up 2 units.
Answer: reflected over x-axis, shifted right 3 units and up 2 units
9) 𝑓(𝑥) = −√𝑥 − 3 − 1: The negative in front of the square root will reflect over the x-axis. The -3
under the square root will shift right 3 units and the -1 will move it down 1 unit.
Answer: reflected over x-axis, shifted right 3 units and down 1 unit
11) 𝑓(𝑥) = √−𝑥 + 3: the – in front of the x will reflect the graph over the y-axis. The +3 will shift up 3
units.
Answer: reflect over y-axis and shift up 3 units
13) 𝑓(𝑥) = √−𝑥 + 1 : the – in front of the x will reflect the graph over the y-axis. The +1 will shift left
1 unit
Answer: reflect over y-axis and left one unit
15) f(x) = −√−𝑥 − 1 the – in front of the square root will reflect over the x-axis, the – in front of the x
will reflect the graph over the y-axis. The -1 will shift down 1 unit.
Answer: reflect over x-axis and y-axis and shift down 1 unit
17) Write the function whose graph has the same shape as the graph of f(x) = x2 but is shifted to the
right 3 units.
I need a -3 under the parenthesis to do this:
Answer g(x) = (x-3)2 (I named the function g so that I didn’t need to write an elaborate name using the
f name. I could also have named the function f(x-3)
19) Write the function whose graph has the same shape as the graph of f(x) = x2 but is shifted to the
right 3 units and up 2 units.
I need a -3 in the parenthesis to move right 3 units and a +2 after the parenthesis to move up 2 units.
Answer g(x) = (x-3)2 + 2
21) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the right 5 units and down 2 units.
I need a – in front of a parenthesis to reflect over x-axis. I need a -5 inside parenthesis to shift right 5
and a (– 2) after parenthesis to move down 2 units.
Answer g(x) = -(x-5)2 – 2
23) Write the function whose graph has the same shape as the graph of f(x) = √𝑥 + 3 but is reflected
over the y-axis.
I need to change the x to (-x) to reflect over y-axis
Answer: g(x) = √−𝑥 + 3
25) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the left 4 units.
I need a – in front parenthesis to reflect over x-axis and a +4 inside parenthesis to move 4 left
Answer: g(x) = -(x+4)2
27) Write the function whose graph has the same shape as the graph of f(x) = x2 but is reflected over the
x-axis and is shifted to the left 5 units and up 2 units.
I need a – in front of the parenthesis to reflect over x-axis, a +5 inside parenthesis to move left 5 and a
+2 after parenthesis to move up 2.
Answer g(x) = -(x+5)2 + 2
29) I know the equation needs an absolute value because of the shape of the graph. The graph is
upside down, so I need a negative in front of the absolute value symbol. The graph is shifted right 2 and
I need a -2 under the absolute value for this. The graph is shifted up 4 and I need a +4 after the absolute
value for this.
Answer: 𝑓(𝑥) = −|𝑥 − 2| + 4
31) The equation needs a square as it is a parabola. The graph is shifted right 2 units and I will need a (2) inside the parenthesis for this. The graph is shifted up 1 unit and I will need a +1 after the parenthesis
for this.
Answer: f(x) = (x-2)2 + 1
33) The equation needs a square as it is a parabola. There needs to be a – in front of the parenthesis
since it is flipped over, a -3 inside the parenthesis will move it right 3 units and a -4 after the parenthesis
will move it down 4.
Answer: f(x) = -(x-3)2 – 4
35a) we say the graph is compressed horizontally,
the graph of f(x) = |𝑥| is the solid line.
35b) we say the graph is compressed horizontally.
the graph of f(x) = |𝑥| is the solid line.
Section 3.5
35c) we say the graph is stretched
horizontally. the graph of f(x) = |𝑥| is the solid
line.
37a) we say the graph is stretched vertically. the
graph of f(x) = |𝑥| is the solid line
35d) we say the graph is stretched horizontally. the
graph of f(x) = |𝑥| is the solid line.
37b) we say the graph is stretched vertically. the
graph of f(x) = |𝑥| is the solid line
Section 3.5
37c) we say the graph is compressed vertically.
the graph of f(x) = |𝑥| is the solid line. The graph
is also reflected over the x - axis
37d) we say the graph is compressed vertically. the
graph of f(x) = |𝑥| is the solid line