Chapter 7

7.2 Graphs of Cosine Functions

Graph of the Coine Function y = cos x

fig 7.2-1

Domain: The set ℝ of all real numbers.       Range: {y| -1 ≤ y ≤ 1}
$$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2},0), (\dfrac{3\pi}{2} , 0)\\ \text{maximum point:} (0,1), (2\pi, 1) \quad \text{minimum point:} (\pi,-1) \end{cases} } $$ Note that sin (x + π⁄2) = cos x,
$$y = \text{sin}\, x \xrightarrow[- \dfrac{\pi}{2} \, \text{units}] {\text{horizontal translation}} \, y = \text{cos}\, x $$

Graph of the Cosine Function y = a cos bx, a > 0, b > 0

From the graph of y = cos x, the graph of y = a cos bx can be obtained as

$$\scriptsize{y = a \, \text{sin}\, bx\, \xrightarrow[\text{horizontal scaling, scale factor}\, \tiny{\dfrac{1}{b}}] {\text{vertical scaling, scale factor}\, a} \, y = a \, \text{cos}\, bx}$$ $$\scriptsize{(x,y) \rarr \, (\dfrac{x}{b}, ay) }$$

fig 7.2-2

$$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2b},0), (\dfrac{3\pi}{2b} , 0)\\ \text{maximum point:} (0,a), (\dfrac{2\pi}{b}, a) \quad \text{minimum point:} (\dfrac{\pi}{b},-a) \end{cases} } $$

Graph of the Cosine Function y = -a cos bx, a > 0, b > 0

$$y = a\, \text{cos}\, bx\, \xrightarrow{\text{Reflection on the}\, x\text{-axis}}\, y = -a\, \text{cos}\, bx $$
Example 5.
Draw the graphs of y = 1⁄2 cos 2x and y = - 1⁄2 cos 2x
Solution
fig 7.2-3

Note that y = a cos (-b)x = a cos bx, so no need to consider b < 0.

Graph of the Cosine Function y = a cos b(x - h) + k   , a > 0, b > 0

$$\small{y = a \, \text{cos}\, bx\, \xrightarrow[\text{vertical translation}\, k\, \text{units}] {\text{horizontal translation}\, h\, \text{units}} \, y}$$ $$\small{= a \, \text{cos} \, b(x - h) + k }$$

fig 7.2-4

y = cos x → y = a cos bx → y = a cos b(x - h) + k
(x, y) → (x⁄b, ay) → ( x⁄b + h, ay + k)

Five key points: points on the midline : (π⁄2b + h, k, (3π⁄2b + h, k)
maximum points: (h, a + k), (2π⁄b + h, a+k)     minimum point: (π⁄b + h, -a+k)

Example 6.
Draw the graph of y = 2 cos π⁄3 (x + 1) + 3 and y = -2 cos π⁄3 (x + 1) + 3.
Solution
Midline: y = 3,
Amplitude: 2,
Period: 6
Five key points for y = 2 cos π⁄3 (x + 1) + 3: (-1, 5), (0.5, 3), (2, 1), (3.5, 3), (5, 5)
Five key points for y = - 2 cos π⁄3(x + 1) + 3: (-1, 1), (0.5, 3), (2, 5), (3.5, 3), (5, 1)
fig 7.2-5

Exercise 7.2
1. From the graph of y = cos x, draw step-by-step transformation graphs to get the graph of y = -3 cos π⁄2 x.
Solution
$$y = \text{cos}\, x \xrightarrow[\text{scale factor 3}] {\text{vertical scaling}} \, y $$ $$= 3\, \text{cos}\, x \, \xrightarrow[\text{scale factor}\, \tiny{\dfrac{1}{\dfrac{\pi}{2}}\, \tiny{-\dfrac{2}{\pi}} }]{\text{horizontal scaling}} y $$ $$= 3\, \text{cos}\, \dfrac{\pi}{2}\, x \, \xrightarrow[\text{on the}\, x{-axis}]{\text{Reflection}} y $$ = -3 cos π⁄2 x
(x, y) ⟶ (x, 3y) ⟶ (2x⁄π, 3y) ⟶ (2x⁄π, -3y)
(0, 1) ⟶ (0, 3) ⟶ (0, 3) ⟶ (0,-3)
(π⁄2, 0) ⟶ (π⁄2, 0) ⟶ (1, 0) ⟶ (1, 0)
(π, -1) ⟶ (π, -3) ⟶ (2, -3) ⟶ (2, 3)
(3π⁄2, 0) ⟶ (3π⁄2, 0) ⟶ (3, 0) ⟶ (3, 0)
(2π, 1) ⟶ (2π, 3) ⟶ (4, 3) ⟶ (4, -3)
fig 7.1-d1

fig 7.1-d2

fig 7.1-d3

2. Draw the graph of (a) y = 1⁄2 cos x
(b) y = cos 4x.
Solution
(a) y = 1⁄2 cos x
Comparing with y = a cos bx,
a = 1⁄2,   b = 1
$$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2b} ,0), (\dfrac{3\pi}{2b} , 0)\\ \text{maximum point:} (0,a), (\dfrac{2\pi}{b},a) \quad \text{minimum point:} (\dfrac{\pi}{b},-a) \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2\sdot 1} ,0), (\dfrac{3\pi}{2\sdot 1} , 0)\\ \text{maximum point:} (0,\dfrac{1}{2}), (\dfrac{2\pi}{1},\dfrac{1}{2}) \quad \text{minimum point:} (\dfrac{\pi}{1},- \dfrac{1}{2}) \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2} ,0), (\dfrac{3\pi}{2} , 0)\\ \text{maximum point:} (0,\dfrac{1}{2}), (2\pi,\dfrac{1}{2}) \quad \text{minimum point:} (\pi,- \dfrac{1}{2}) \end{cases} } $$
Domain: R
Range: {y | - 1⁄2 ≤ y ≤ 1⁄2}
Period: 2π
Amplitude: 1⁄2
x-intercept: (π⁄2, 0), (3π⁄2, 0)
Maximum point: (0, 1⁄2), (2π 1⁄2)
Minimum point: (π, - 1⁄2)
fig 7.1-d4

(b) y = cos 4x
Comparing with y = a cos bx,
a = 1,   b = 4
$$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2b} ,0), (\dfrac{3\pi}{2b} , 0)\\ \text{maximum point:} (0,a), (\dfrac{2\pi}{b},a) \quad \text{minimum point:} (\dfrac{\pi}{b},-a) \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2\sdot 4} ,0), (\dfrac{3\pi}{2\sdot 4} , 0)\\ \text{maximum point:} (0,1), (\dfrac{2\pi}{4},1) \quad \text{minimum point:} (\dfrac{\pi}{4},-1) \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{8} ,0), (\dfrac{3\pi}{8} , 0)\\ \text{maximum point:} (0,1), (\dfrac{\pi}{2},1) \quad \text{minimum point:} (\dfrac{\pi}{4}, -1) \end{cases} } $$
fig 7.1-d5


3. Draw the graph of (a) y = 2 cos π⁄4 x
(b) y = -sin πx.
Solution
(a) y = π⁄4 x
Comparing with y = a coss bx,
a = 2,   b = π⁄4
$$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2b} ,0), (\dfrac{3\pi}{2b} , 0)\\ \text{maximum point:} (0,a), (\dfrac{2\pi}{b},a) \quad \text{minimum point:} (\dfrac{\pi}{b},-a) \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2\sdot \dfrac{\pi}{4}} ,0), (\dfrac{3\pi}{2\sdot \dfrac{\pi}{4}} , 0)\\ \text{maximum point:} (0, 2), (\dfrac{2\pi}{\dfrac{\pi}{4}}, 2) \quad \text{minimum point:} (\dfrac{\pi}{\dfrac{\pi}{4}},-2) \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (2 ,0), (6, 0)\\ \text{maximum point:} (0, 2), (8, 2) \quad \text{minimum point:} (4, -2) \end{cases} } $$
fig 7.1-d6

(b) y = -cos πx
Comparing with y = a cos bx,
a = -1,   b = π
$$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2b} ,0), (\dfrac{3\pi}{2b} , 0)\\ \text{maximum point:} (0,a), (\dfrac{2\pi}{b},a) \quad \text{minimum point:} (\dfrac{\pi}{b},-a) \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{\pi}{2\pi} ,0), (\dfrac{3\pi}{2\pi} , 0)\\ \text{maximum point:} (0, -1), (\dfrac{2\pi}{\pi}, -1) \quad \text{minimum point:} (\dfrac{\pi}{\dfrac{\pi}{\pi}},-\tiny{(}-1\tiny{)} \scriptsize{)} \end{cases} } $$ $$\scriptsize{ \text{Five key points:} \begin{cases}x\text{-intercepts:} (\dfrac{1}{2} ,0), (\dfrac{3}{2}, 0)\\ \text{maximum point:} (0, -1), (2, -1) \quad \text{minimum point:} (1, 1) \end{cases} } $$
fig 7.1-d7

4. Draw the graph of (a) y = 2 cos π(x-2) + 1
(b) y = -2 cos 1⁄2(x+1) + 2.
Solution
(a) y = 2 cos π(x-2) + 1
Comparing with y = a cos b(x-h) + k,
a = 2,   b = π,   h = 2,   k = 1
Key points on the midline y = k
(h, k), (π⁄b+h, k), (2π⁄b+h, k)
Maximum and minimum points
(π⁄2b+h, a+k), (3π⁄2b+h, -a+k)

Key points on the midline y = k
(2, 1), (π⁄π+2, 1), (2π⁄π+2, 1)
Maximum and minimum points
(π⁄2π+2, 2+1), (3π⁄2π+2, -2+1)

 Key points on the midline y = k = 1
(2, 1), (3, 1), (4, 1)
Maximum and minimum points
(5⁄2, 3), (7⁄2, -1)

fig 7.1-d8


(b) y = -2 cos 1⁄2(x+1) + 2.
Comparing with y = a cos b(x-h) + k,
a = -2,   b = 1⁄2,   -h = 1 or h = -1,   k = 2
Key points on the midline y = k
(h, k), (π⁄b+h, k), (2π⁄b+h, k)
Maximum and minimum points
(π⁄2b+h, a+k), (3π⁄2b+h, -a+k)

Key points on the midline y = k
(-1, 2), (π⁄ 1⁄2 +(-1), 2), (2π⁄ 1&frasl2 +(-1), 2)
Maximum and minimum points
(π⁄2 ⋅ (1&frasl2) -1, -2+2), (3π⁄2 (1&frasl2) -1, -(-2)+2)

 Key points on the midline y = k
(-1, 2), (2π-1, 2), (4π-1, 2)
Maximum and minimum points
(π-1, 0), (3π-1, 4)

fig 7.1-d9


5. Show that y = cos bx is an even function.
Solution
Let f(x) = y = a cos bx
f(-x) = a cos b(-x)
= a cos (-bx)
= a cos bx     (∵ cos (-θ) = cos (θ) )
∴ y = a cos bx is an even function.