Year 10 Extended · Unit 3 Functions, quadratics and transformations · Lesson 6 of 7

Changing a rule to move its picture

Adding to the input moves a graph sideways, adding to the output moves it up or down, and multiplying either one stretches or squashes it. What is inside the bracket and what is outside it do opposite kinds of job.

FunctionsGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: reflect the point (3, −5) in the x-axis.
    Answer(3, 5). A reflection in the x-axis changes the sign of the y-coordinate.
  2. From this unit, Lesson 3: the vertex of y = (x − 7)² + 2 is (h, k). What is h?
    Answer7. The bracket is (x − h), so h = 7.
  3. From Year 9: what is the image of (4, −1) when it is translated 3 left and 2 up?
    Answer(1, 1). Subtract 3 from x and add 2 to y.
Words used in this lesson
translation
sliding every point the same distance in the same direction. For example: y = f(x) + 9 moves the graph 9 up.
reflection
flipping a graph in a mirror line. For example: y = −f(x) in the x-axis.
dilation
stretching or squashing by a scale factor. For example: y = 10f(x).
image
the point or graph after a transformation. For example: (2, 8) is the image of (2, 3) under y = f(x) + 5.
scale factor
the number a length is multiplied by. For example: 1 ÷ 2 for f(2x).
parent graph
the original graph before it is transformed. For example: y = f(x).

The ridge in the game

A game designer in Seoul draws a ridge-shaped platform with the rule y = f(x). She wants the same ridge again, further along the level, and once more upside down on the ceiling. She does not want to draw the shape three times; she wants to change the rule.

Predict first: to see the same ridge 3 units further right, should the rule be f(x + 3) or f(x − 3)? Then use the figure: move, flip and stretch the ridge, and follow one corner.

0
0
1
1
2
y = f(x)new rule
(−2, 3) → (−2, 3)followed corner

y = f(x): the point (−2, 3) goes to (−2, 3).

Show me the steps
  1. Slide h to 3 and leave the rest. Every point moves 3 to the right: the corner (−2, 3) goes to (1, 3). The rule is y = f(x − 3), and the minus sign inside the bracket is the opposite of the way the graph moves.
  2. Put h back to 0 and slide k to −2. Every point moves 2 down: (−2, 3) goes to (−2, 1). The rule is y = f(x) − 2, and this time the sign says exactly what happens.
  3. Press Flip in the x-axis: the rule is y = −f(x) and (−2, 3) goes to (−2, −3). Press it again, then Flip in the y-axis: the rule is y = f(−x) and (−2, 3) goes to (2, 3).
  4. With the flips off, slide a to 2: y = 2f(x) and (−2, 3) goes to (−2, 6), so heights double and x-coordinates do not change. Then put a back to 1 and slide b to 2: y = f(2x) and (−2, 3) goes to (−1, 3), a squash towards the y-axis.

Outside the bracket acts on y, and does what it says. y = f(x) + k moves the graph up k. y = −f(x) reflects it in the x-axis. y = af(x) multiplies every height by a, a vertical stretch from the x-axis.

Inside the bracket acts on x, and does the opposite. y = f(x − h) moves the graph h to the right, because the output at x is what f gave at x − h. y = f(−x) reflects it in the y-axis. y = f(bx) divides every x-coordinate by b, a squash towards the y-axis by factor 1 ÷ b.

These rules do not depend on f: they work for lines, parabolas, roots, or a zigzag.

f(x + 6) moves left 6, and f(3x) is a squash, not a stretch. To see why the first moves left, ask which x gives the old input: x + 6 equals a given old input when x is 6 less. In f(3x) the input grows 3 times as fast, so features arrive 3 times sooner: every x-coordinate is divided by 3.

Worked examples

1. A graph has a peak at (3, 6). Where is the peak of y = f(x + 2) − 4?

  1. f(x + 2) moves the graph 2 to the left: (3, 6) goes to (1, 6).
  2. The −4 outside moves it 4 down: (1, 6) goes to (1, 2).
  3. The peak is at (1, 2).

2. A graph passes through (4, −2). Find the image under y = f(−x) + 1.

  1. f(−x) reflects in the y-axis: the x-coordinate changes sign, (4, −2) goes to (−4, −2).
  2. The +1 outside moves every point up 1: (−4, −2) goes to (−4, −1).
  3. The image is (−4, −1).

3. A graph has a zero at x = 6. Where is the zero of y = f(3x)?

  1. The zero is where 3x equals the old zero: 3x = 6.
  2. So x = 6 ÷ 3 = 2: the x-coordinate is divided by 3.
  3. Check with a squash: every x-coordinate is one third as far from the y-axis, and 6 is now 2.

Your go. The first step is given. A graph has a lowest point at (−3, 5). Find the lowest point of y = 2f(x − 4) + 1.

  1. The x-coordinate comes only from f(x − 4), which moves the point 4 to the right: −3 + 4 = 1.
  2. Now apply the 2 (outside, on y) and then the +1 to the y-coordinate.
Check the answer

The y-coordinate is 2 × 5 + 1 = 11, so the lowest point is (1, 11). The stretch comes before the +1, because it multiplies f(x − 4) only.

With a GDC (optional)

A GDC can show a transformation. Enter y1 = x² and y2 = (x + 2)² − 1 and graph both: y2 is the first graph moved 2 to the left and 1 down. Try y3 = −y1 and y4 = (−x)² too.

A single sketch can look right while the rule is wrong, so read the rule first and use the GDC to confirm it. Everything here can be done by hand.

Check before you go on

1. The graph of y = f(x) passes through (6, 3). Which point must be on the graph of y = f(x − 2) + 4?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The graph of y = f(x) has its lowest point at (−1, −6). The graph of y = f(x + 4) has its lowest point at (p, q). Find p.

3. The point (−3, 7) is on y = f(x). It maps to (p, q) on y = −f(−x). Work out p + q.

Level 2 · Building

4. The graph of y = f(x) has a zero at x = 5. The graph of y = f(2x) has a zero at x = ? Give your answer as a decimal.

5. The greatest value of f(x) is 4, at x = 1. What is the greatest value of 3f(x − 2) − 5?

Level 3 · Stretch

6. The graph of y = x² is reflected in the x-axis, then translated 3 to the right, then translated 8 up. At what value of y does the new graph meet the y-axis?

7. The graph of y = f(x) has zeros at x = −2 and x = 6. What is the larger zero of y = f(3 − x)?

Something to think about

Set h = 4 and press Flip in the y-axis. Corner 2, (−2, 3), goes to (6, 3), so the figure flips first and then moves 4 right: f(−(x − 4)). Now work out by hand where (−2, 3) goes if you move it 4 right first and then flip in the y-axis. Is it the same point? Write the rule for that order.

One way to think about it

Moving first takes (−2, 3) to (2, 3), and the flip then takes it to (−2, 3), not (6, 3). The rule for that order is f(−x − 4) = f(−(x + 4)): flipping after the move turns the 4 right into 4 left. The two orders give different graphs, and that is the subject of the next lesson.

Exit ticket

8. Which description matches y = −f(x + 1)?