Graphs and Their Transformations

Try this

Drag only $φ$ . The whole curve slides left and right, and the red dot (the image of $(0, 0)$ , the start of $sin x$ ) moves with it. Where does it stop?
Set $ω$ to $1$ and drag $φ$ ; then set it to $3$ and drag again. With $φ$ dragged to $\frac{π}{2}$ both times, does the red dot travel the same distance?
Drag only $ω$ . How does the period reading change? Do neighboring peaks move closer together or further apart?
Make $A$ negative. What happens to the curve? Where does the blue dot (the old highest point) go?
Drag $k$ : only the range changes, the period and the shift do not.

In one sentence: in $y = A sin (ω x + φ) + k$ , $A$ stretches up and down, $k$ moves up and down, $ω$ stretches left and right, and the left–right shift is

- \frac{φ}{ω} (the x -coordinate of the red dot), T = \frac{2 π}{∣ ω ∣}, range [k - ∣ A ∣, k + ∣ A ∣] .

The $x$ with $ω x + φ = 0$ is the “start”. Whenever you study a function like this, first treat $ω x + φ$ as one block: it is the most useful move in this chapter.

Predict first

How can the graph of $y = sin (2 x + \frac{π}{3})$ be obtained from the graph of $y = sin 2 x$ ?

shift left by

\frac{π}{3}

shift left by

\frac{π}{6}

shift right by

\frac{π}{3}

shift right by

\frac{π}{6}

Choose an answer first; then you can see the result.

Why: build it one step at a time

Starting from $y = sin x$ , we build $y = 2 sin (2 x + \frac{π}{3}) + 1$ . The thick pale green line is the target; the dashed line is how things looked one step earlier.

From y = sin x to y = 2sin(2x + π/3) + 1

Step 1 / 6

Start
$y = sin x$ . The red dot is the start $(0, 0)$ and the blue dot is the highest point $(\frac{π}{2}, 1)$ . At every step below, watch where these two points go.

$y = sin x$
Shift by φ first
Then squeeze sideways
Stretch vertically, then lift
Other order: stretch first
What if we still shift by π/3?

If you change the order

Break it

Counterexample lab: remove one condition

Conclusion: starting from $y = sin x$ , a shift by $φ$ gives $y = sin (ω x + φ)$

Shift by $\frac{π}{2}$ first, then shrink the $x$ -coordinates to $\frac{1}{3}$ : click through to the last step and the red curve matches the target.

Practice

Exercise 1Basic

The least positive period of $y = 3 sin (\frac{x}{2} - \frac{π}{4})$ is $k π$ . Find $k$ .

Exercise 2Basic

Shifting the graph of $y = sin 2 x$ right by $a$ units gives the graph of $y = sin (2 x - \frac{π}{3})$ . If $a = mπ$ ( $0 < a < π$ ), find $m$ .

Exercise 3Advanced

Shrink the $x$ -coordinate of every point on the graph of $y = sin x$ to $\frac{1}{3}$ of its value ( $y$ -coordinates unchanged), then shift left by $\frac{π}{4}$ units. The result is the graph of $y = sin (3 x + φ)$ ( $0 < φ < π$ ). If $φ = k π$ , find $k$ .

Exercise 4Challenge

The graph of $f (x) = A sin (ω x + φ)$ ( $A > 0, ω > 0, ∣ φ ∣ < \frac{π}{2}$ ) has a highest point at $(\frac{π}{12}, 2)$ , and the nearest zero to its right is $x = \frac{π}{3}$ . If $φ = k π$ , find $k$ .

Try this

Why: build it one step at a time

From y = sin x to y = 2sin(2x + π/3) + 1

If you change the order

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Reading the shift straight off the constant in the brackets

Thinking a larger ω makes the graph wider